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AMERICAN MATHEMATICAL SOCIETY
COLLOQUIUM PUBLICATIONS
VOLUME XVIII
THE CALCULUS OF VARIATIONS
IN THE LARGE
BY
MARSTON MORSE
PUBLISHED BY THE
AMERICAN MATHEMATICAL SOCIETY
501 West 110th Street, New York
1934
Photo-Lithoprint Reproduction
EDWARDS BROTHERS, INC.
Lithoprinters
ANN ARBOR, MICHIGAN
1947
FOREWORD
For several years the research of the writer has been oriented by a conception
of what might be termed macro-analysis. It seems probable to the author that
many of the objectively important problems in mathematical physics, geometry,
and analysis cannot be solved without radical additions to the methods of what
is nowr strictly regarded as pure analysis. Any problem which is non-linear in
character, which involves more than one coordinate system or more than one
variable, or whose structure is initially defined in the large, is likely to require
considerations of topology and group theory in order to arrive at its meaning
and its solution. In the solution of such problems classical analysis will fre¬
quently appear as an instrument in the small, integrated over the whole problem
with the aid of group theory or topology. Such conceptions are not due to the
author. It will be sufficient to say that Henri Poincare was among the first to
have a conscious theory of macro-analysis, and of all mathematicians was
doubtless the one who most effectively put such a theory into practice.
The principal contribution of the author has been first to give an analysis in
the large of a function / of m variables, and then to extend this analysis to
functionals. The functionals chosen have been those of the Calculus of Varia¬
tions. Although there are indications that further deep extensions to other
functionals exist, such extensions are beyond the scope of these Lectures.
Whereas the analogies between the theory of linear and quadratic forms and the
theory of functionals have been well recognized since the work of Hilbert, the
analogies in the large between functions and functionals here presented have not
been so recognized, and the nature of the development of such analogies in
many aspects has been most difficult.
The first four chapters of these Lectures deal with the theory in the small.
They are concerned with the analogue for functionals, of the index of a critical
point of the function /. Conjugate points, focal points, characteristic roots,
the Poincar6 rotation number, and the index of concavity of closed extremals
are among the entities which serve to evaluate the index of a critical extremal,
and which are unified by the theory of this index.
Chapter IV goes beyond the needs of the theory in the large in developing
separation, comparison, and oscillation theorems in n-space. The most general
algebraic form of linear, self-adjoint boundary conditions associated with the
usual Jacobi differential equations is exposed in a parametric form in which
only those coefficients appear which are arbitrary. The theory is sufficiently
refined to specialize into a definite improvement upon the oscillation theorems of
B6cher [2] and Ettlinger [1, 2] in the 2-dimensional periodic case. Among other
theorems, a necessary and sufficient condition for the existence of infinitely
many characteristic roots in our self-adjoint boundary problems is established.
iii
iv
FOREWORD
Except for a theorem on the order of vanishing of the determinant of a conjugate
family, most of the work of the first four chapters can be readily extended to the
Bolza form of the Lagrange problem if the proper assumptions as to “normalcy”
are made.
Chapter V presents the general boundary problem in the large. It starts with
a macroscopic definition of a Riemannian manifold 12. The functional and
boundary conditions on 12 are defined in parametric form, and in the large. A
first problem which is solved concerns the invariantive or tensor definition of the
indices of the preceding chapters. This aspect of the theory will be of interest to
differential geometers. Chapter V treats the general accessory boundary
problem in a way which is independent of the local coordinate systems employed.
The author believes that this is the first general treatment of this character.
Chapter VI develops the theory of the critical points of a function of m
variables in a manner which seems best adapted to an extension to the case of
functionals. The analogous treatment for the case of functionals requires the
development of the topology of the function space defined by a given boundary
problem. For problems for which the end points are always distinct, the func¬
tion space can be treated as in Chapter VII. The theory of the closed extremal
in Chapter VIII requires a new approach to the topology of the corresponding
function space. In particular homologies which are not defined by bounding
are used here, and subgroups of substitutions of q points play an important r61e.
Chapter IX presents a solution of the Poincar6 continuation problem which
arose from Poincare's study of Celestial Mechanics, Poincare [2]. With Poincare
this problem reduced to the question of the existence and analytic continuation
of a closed geodesic on a convex surface as the surface was varied analytically.
Poincare started with the principal ellipses on an ellipsoid. The validity of his
reasoning has been questioned. In Chapter IX explicit objections are presented.
The present writer enlarges the Poincare continuation problem to mean the
problem of finding those numerical invariants of critical sets of closed extremals,
the possession of which is a guarantee of the continued existence and analytic
variation of critical sets possessing the given numerical invariants as the basic
Riemannian manifold is varied analytically. This theory is applied to show that
on an m-ellipsoid with unequal axes the principal ellipses vary analytically into
critical sets of geodesics with the same numerical invariants, as the w-ellipsoid is
varied analytically through a 1-parameter family of closed manifolds.
The author takes occasion here to acknowledge his principal sources. First
of all the author wishes to acknowledge his indebtedness to his colleague,
Professor George D. Birkhoff, whose minimax principle, Birkhoff [1], was the
original stimulus of the present investigations, and whose transformation theory
of dynamics, though logically less closely related to these Lectures, has by virtue
of its broad aims and accomplishments proved no less inspiring. The author's
knowledge of the classical theory has been acquired largely from the treatises of
Bolza and Hadamard, and from the works of Bliss whose papers on the n-
dimensional theory have been particularly useful. In topology the author has
FOREWORD
V
been fortunate in having the contemporary work of Veblen, Alexander, and
Lefschetz to follow, and to have had their papers always at his disposal.
The bibliography at the end of the Lectures is not intended to be complete,
but merely to list recent papers used by the author, or papers which may be
regarded as related to the work of the author.
The author acknowledges the generous aid furnished him by the Milton Fund
of Harvard University for the preparation of the manuscript. Dr. S. B. Myers
and Dr. A. W. Tucker have been kind enough to read parts of the text and to
offer valuable suggestions. Dr. Nancy Cole has greatly assisted both in the
reading and in the preparation of the manuscript.
The author extends his thanks to the American Mathematical Society and to
its officers for their invitation to present and publish these Lectures.
Cambridge, Massachusetts.
TABLE OF CONTENTS
SECTION PAGE
Foreword . iii
Chapter I
THE FIXED END POINT PROBLEM IN NON-PARAMETRIC FORM
1. The Euler equations . 1
2. The existence of extremals . 4
3. The necessary conditions of Weierstrass and Legendre . 5
4. The Jacobi condition . 7
5. Conjugate points . 9
6. The Hilbert integral . 13
7. Sufficient conditions . 15
Chapter II
GENERAL END CONDITIONS
1. The end conditions . 18
2. The transversality condition . 20
3. The second variation . 21
4. The accessory boundary problem . 24
5. The necessary condition on the characteristic roots . 26
6. The non-tangency hypothesis . 28
7. The form Q(uf \) . 30
8. Sufficient conditions . 33
Chapter III
THE INDEX FORM
1. Definition of the index form . 37
2. Properties of the index form . 42
3. Conjugate families . 46
4. Necessary conditions, one end point variable . 49
5. Focal points . ... ... 51
6. The index of g in terms of focal points 55
7. Certain lemmas on quadratic forms . 61
8. Two end manifolds . 64
9. Periodic extremals, a necessary condition . 70
10. The order of concavity . 71
11. The index of a periodic extremal . 74
Chapter IV
SELF-ADJOINT SYSTEMS
1. Self-adjoint differential equations . 80
2. A representation of self-adjoint boundary conditions . 83
3. Boundary problems involving a parameter . 89
4. Comparison of problems with different boundary conditions . 92
5. A general oscillation theorem . 95
vii
viu TABLE OF CONTENTS
6. The existence of characteristic roots . 97
7. Comparison of problems possessing different forms u> . 99
8. Boundary conditions at one end alone . 102
Chapter V
THE FUNCTIONAL ON A RIEMANNIAN SPACE
L A Riemannian space in the large . 107
2. Basic tensors . Ill
3. The necessary conditions of Euler, Weierstrass, and Legendre . 113
4. Extremals . 115
5. Conjugate points . 117
6. The Hilbert integral . 119
7. Sufficiency theorems . 120
8. The Jacobi equations in tensor form . 122
9. The general end conditions . 126
10. The second variation . 126
11. The accessory problem in tensor form . 127
12. The non-tangency condition . 131
13. Characteristic solutions in tensor form . 133
14. The general index form . 137
15. The case of end manifolds . 138
Chapter VI
THE CRITICAL SETS OF FUNCTIONS
1. The non-degenerate case . 142
2. The problem of equivalence . 146
3. Cycles neighboring <r . 151
4. Neighborhood functions . 152
5. The determination of spannable and critical sets . 156
6. Classification of cycles . 158
7. The type numbers of a critical set . 165
8. Justification of the count of equivalent critical points . 175
9. Normals from a point to a manifold . 179
10. Symmetric squares of manifolds . 181
11. Critical chords of manifolds . 183
Chapter VII
THE BOUNDARY PROBLEM IN THE LARGE
1. The functional domain 12 . 193
2. The function J(tc) . 196
3. The domain J(t) <6 . 200
4. Restricted domains on 12 . 205
5. The /-distance between restricted curves . 208
6. Cycles on 12 neighboring a critical set w . 212
7. The space 2 of /-normal points . 213
8. Theorem 6.1 . 216
9. Cycles on the domains / < b and / < a . 220
10. The existence of critical extremals . 221
11. The non-degenerate critical extremal . 226
12. The non-degenerate problem . 230
13. The fixed end point problem . 234
TABLE OF CONTENTS ix
14. The one variable end point problem . 240
15. The two point functional connectivities of an w-sphere . 244
Chapter VIII
CLOSED EXTREMALS
1. The complexes K , Kp, and IP . 250
2. The infinite space 12 . 253
3. Critical sets of extremals . 256
4. The domain IP . 258
5. Critical sets on IP . 261
6. Critical sets on 12 . 264
7. The extension of a chain on IP . 273
8. The r-fold join of a cycle . 277
9. Finiteness of the basic maximal sets . 285
10. Numerical invariants of a closed extremal g. . . 288
11. The non-degenerate closed extremal . 291
12. Metrics with elementary arcs . 297
Chapter IX
SOLUTION OF THE POINCAR& CONTINUATION PROBLEM
1. Regular submanifolds of Rp . 307
2. Geodesics on ra-ellipsoids . 312
3. The indices of the ellipses qh . 316
4. The exclusiveness of the closed geodesics gra . 319
5. The linking cycles A12 (a) . 323
6. Symmetric chains and cycles . 326
7. The linking cycles \r%j(a) . 333
8. The circular connectivities of the m-sphere . 346
9. Topologically related closed extremals . 350
10. Continuation theorems . 354
Bibliography . 359
Index . 367
CHAPTER I
THE FIXED END POINT PROBLEM IN NON-PAR AMETRIC FORM
The original plan of these Lectures was to start with a treatment of the
problem under general end conditions. However the lack of a complete treat¬
ment in book form in English of the classical ^dimensional theory made it seem
desirable to depart from this plan to the extent of giving an introductory chapter
on the fixed end point problem. This chapter is an exposition of classical results
treated for the most part by classical methods F ree use is made of the works of
Bliss, Bolza, and Hadamard.
The Euler equations
1. Let it be recalled that a function F of n variables ( w ) is said to be of class
Cmj rn ^ 0, in the variables ( w ) on a domain S, if F is continuous on S, together
with all of its partial derivatives up to and including those of the mth order. A
function y(: r) of a single variable x is said to be of class Dm, rn > 0, on an interval
a i ^ x g b, if y{x) is continuous on the interval, and if the interval can be divided
into a finite set of subintervals on the closure of each of which y(x) is of class
Cm. The function y(x) will be said to be of class D° on the interval a g x ^ b, if
this interval can be divided into a finite set of subintervals on the interior of
each of which y(x) is of class C° and at the ends of which y(x) possesses finite right
and left hand limits.
Let
(*, Vh * * * , Vn) = Or, y)
be the rectangular coordinates of a point (a, y) in a euclidean space of (n + 1)
dimensions. Let R be an open region in the space (x} y). We shall consider a
function
/(*, 2/i, • • • , 2 /», Ph ’ ‘ * , Vn) = /Or, V, V)
such that Jy.9fPi and/ are of class C2 for (x, y) on R and for (p) unrestricted.
Let g be a curve in the region R of the form
(1.1) y% = Vi (x) (i = 1, • • * , n)
for x on the interval (a1, a2),
(1.2) a] ^ x g a2
where the functions y%{x) are of class Dl on the interval (1.2). We term g a curve
of class Dl.
In deriving the Euler equations we shall admit curves which have the following
1
THE FIXED END POINT PROBLEM
2
[I]
properties. They are of class Dl on the interval (1.2), and join the end points
of g on R. We shall consider the integral
J = Jol f(x> y*> ■ ■ ■ » y*’ »»>•••> y'n) dx
along these admissible curves and prove the following theorem.
Theorem 1.1. In order that the curve g afford a minimum to J relative to neigh¬
boring admissible curves it is necessary that g satisfy the conditions
(1.3) fp.(x, y(x), y'(x)) = / fv.(x, y(x), y'{x)) dx + a
J a1
for x on its interval (1.2), and for suitable constants c».
In proving this theorem one considers a family yi = yi(xt e) of admissible
curves of the form
(1.4) yi(x, e) = yi(x) + ein(x) (i = 1, • • • , n)
where e is a parameter near 6 = 0, rn(x) is of class Dl on the interval (1.2) and
vanishes at a1 and a2. For each value of e near e = 0 we thus obtain a value
J(e) of the integral J. Moreover if g affords a minimum to J relative to neigh¬
boring admissible curves it is necessary that
(1.5) J’( 0) = [' (v'Jl + vJl) dx =0 0 = 1,--, n)
Ja'
where the superscript 0 indicates evaluation along g, that is for
0, y, v ) = 0, »(*). Vi*))-
Here aim elsewhere we follow a convention of tensor analysis, that a repeated
subscript or superscript i indicates a summation with respect to i. The right
member of (1.5) is called the first variation of J along g . It is determined when g
and the "variations” 17, are given.
The terms rj{f°v in (1.5) can be integrated by parts, giving the result
(1.6) J\ 0) = f Vilfl - f fl dx] dx = 0,
the terms outside the integral having vanished since i?<(al) = ^-(a2) = 0. The
theorem will follow from (1.6) once we have proved the Du Bois-Reymond
Lemma.
Lemma 1.1. If <t>(x) is of class D° on (a1, a2) and
(i-7) r 7 ?'(x)<^(x) dx = 0
for all functions tj(x) of class D 1 which vanish at a1 and a2, then 4>(x) is constant on
(a1, a2).
IM
THE EULER EQUATIONS
3
Let c be a constant such that
The function
i:
v(x)
(<t>(x) — c) dx = 0.
=r
— c) dx
is then a function rj(x ) of the type admitted in the lemma. For this function
ri(x)f (1.7) takes the form
0 -
U>{x) — c)<t>(x) dx
-C
(</>(x) - C y dx,
from which it follows that <t>(x) = c. The lemma is thereby proved.
Returning to (1.6) we take all the functions ru(x) identically zero except one,
say rjk(x). According to the lemma we can infer that the coefficient of ij'k in
the integrand in (1.6) must be constant. This is true for fc = 1,2,-*, n.
The theorem is thereby proved.
We state the following modification of Lemma 1. 1 of use in a later chapter.
Lemma 1.2. If <fr(x) and <l>'(x) are of class D° on (a1, a2) and (1.7) holds for all
functions rf(x) of class D 2 which vanish at a1 and a2, then <f>(x) is constant on (a1, a2).
On the basis of this lemma we could prove as above that a curve y> = yi(x) of
class D 2 on (a1, a2) which affords a minimum to J relative to neighboring curves
of class D 2 which join its end points, satisfies (1.3) as before.
We have the following consequences of the theorem.
Each segment of class C 1 of a minimizing curve g must satisfy the Euler
equations
(L8)
d_
dx
/*-/*- 0
(* = !,•••, »)•
Again, at each corner x = c on g, the right hand and left hand limits of fp on g
are equal, that is, on g
d-9)
M" - 0
(i = 1, • • * , n).
These are the Weierstrass-Erdmann corner conditions.
Suppose g is of class C1, satisfies (1.3), and that along g the determinant
(1.10) \fViPj \ ^ 0.
Then g is of class C2 at least (Hilbert). The proof of this statement according to
Mason and Bliss [1] is as follows. The n equations
(1.11) fPi (x, y(x), z) = f f(x,y(x),y'(x))dx + c,
J a1
4
THE FIXED END POINT PROBLEM
[I]
can be regarded as determining n variables as functions of x. They have the
initial solution zt(x) = y { (x) for each x. Upon taking account of (1.10) and
employing the usual implicit function theorems one sees that the solution
Zi(x) must be of class C\ and hence g of class C2. We shall see later that g is of
class C' provided (1.10) holds.
A curve of class Dl satisfying (1.3) is called a discontinuous solution if it
actually possesses a corner. The theory of discontinuous solutions received a
great impetus from the dissertation of Carath<$odory [1]. A bibliography for
this field has been given by L. M. Graves [1]. Graves has also made many
important contributions. Beyond using the Weierstrass-Erdmann corner con¬
ditions we shall not be concerned with discontinuous solutions.
A curve of class C2 satisfying the Euler equations (1.8) will be called an
extremal .
The existence of extremals
2. Suppose we have an extremal g of the form yx — yt(x) with x on the interval
(a1, a 2). To determine the extremals neighboring g it is useful to set
(2.1) Vi = /„. (x, y, p) (i = 1, • • • , n)
and in particular
Vi(x) = fPi (x, y(x), y'(x)).
We term sets
Or, y(x), y'(x), v(x))
sets (xy y, py v ) on g. We similarly define sets ( x , y, p) or sets (x, yy v) on g.
We assume that the condition (1.10) holds along g. It follows that near sets
(Xy y, p} v ) on g the relation (2.1) can be put in the form
(2.2) pi = pi(x, y, v),
where the functions p,(x, y, v) are of class C2 neighboring sets (x, y, v) on g.
The Euler equations are then equivalent to the equations
(2.3)
= fVi (*, Vi pi*, y , *0),
= p<(*, v, ») (* = i» • • • » »»)>
at least as far as extremals are concerned on which the sets (x, yf v) differ suffi¬
ciently little from similar sets on g.
According to the theory of differential equations, equations (2.3) have solu¬
tions of the form
yi = hi(x, x°, y°, v°),
Vi = ki(x, x°, y°, r°)
(i 1, • • • , n) ,
13]
NECESSARY CONDITIONS
5
which take on the values (y°, v° ) when x = x° , and for which the functions on the
right are of class C 2 in their arguments for x on (a1, a2) or a slightly larger interval,
and for (x°, y°, v° ) sufficiently near sets (x, y, v) on g. We now set
(2.4) (x, x°, a, b) = hi (x, x°, a , /p (x°, a, 6)).
We have in z/t(x, x°, a, b) the general solution of the Euler equations neighboring
the solution g. The functions y%(x9 x°, a, b ) are of class C2 in their arguments
for x on (a1, a2) or a slightly larger interval, and for (x°, a, b) sufficiently near
sets (x, y , ?/') on g. Moreover we have
«* = 2/i (x°, x°, a, b),
(2.5)
b< s yix (x°, x°, a, o),
for (x°, a, b) near sets (x, y , ?/') on g. Reference to the second of equations (2.3)
discloses the additional fact that the functions
and hence
hix Or, x\ y°, v°)
yix (x, x°, a, b)
are of class C2 on the domain of their arguments.
The necessary conditions of Weierstrass and Legendre
3. The Weierstrass L-f unction is defined by the equation
E(x, y, p, q) - f{x, y, q ) - fix, y, p) - (q> - p{) fp.(x, y,p) (i = 1, • • • , n).
We shall prove the following theorem (Weierstrass).
Theorem 3.1. If an arc g of class Cl affords a minimum to J relative to all
neighboring curves of class D 1 joining its end points, then
(3.1) E(x, y, y', q) ^ 0,
for (x, y, y') on g and for any set (g).
Let g be represented as previously by yi = yt(x). Let (xl, yl) be any point of <7.
We treat the case where x1 > a1. The case x1 = a1 requires at most obvious
changes.
Let yi = y%{x) be a short arc y of class Cl, which passes through (x1, yl) when
x == x1, and for which y\{xl) == gi} where is arbitrarily prescribed. The
curves defined for a constant by the family
(3.2) yi = I nix, a) = j? - lyi(a) - yi(a)] + Si(x) (i = 1, ■■ ■ , n),
(a - a1)
reduce to g when a = x1, and in general join the initial point of g to the point
Pa = (x, y) = (a, y(a))
6
THE FIXED END POINT PROBLEM
[l J
on y. We are here supposing that a ^ x\ that a1 g x £ a, and that a is taken
near x1.
We now evaluate J along the curve (3.2) leading from the initial point of g to
Pa , and then along the curve 2/» = ydx) from Pa to (xl, y 1). We have
f(<X> y(%) Vx^y ^ "f* /> x, yix), y'ix )) dx.
If g is a minimizing arc, we must have «7'(xl) g 0. Upon setting p* = y\ (a*1)
we find that
J'ix1) = fix1, y\ v ) - fix \ y\ g) +
y%afy - y laz
dx.
If we integrate the terms involving t/,ax by parts, and make use of the fact that g
must satisfy the Euler equations, we find that
J'ix *) = fix1, y\ p ) - fix1, y\ q ) + x') fp.(x, yix, xl), yzix, x1))]
From the identities
Vi(a\ a) e yi(al), yAa, a) & yi(a ),
it follows, upon differentiating with respect to a and putting a = x1, that
Via(a\ xx) =0, pi + yia(x1t xl) = gi.
Using these results we find that
J\xx) = /Or1, y\ p) - f(xl, ylt g) + (gi - pi ) fp.(xx, yx, p).
The theorem follows from the condition Jf(xx) g 0.
If the hypothesis of the theorem is modified by restricting the admissible
curves to those on which ( x , y, y' ) lies sufficiently near (x, y(x)} y'(x)) on g, the
minimum afforded by g is called a weak minimum. The minimum afforded
by g in the theorem is called a strong minimum. For a weak minimum it is
necessary that the condition (3.1) hold in its weak form; that is, for the sets
(x, p, y'j g) of which (x, yf y') is on g and (g) sufficiently near the set (yr) on g.
As a consequence of this weak condition one can derive the following condition,
due in the plane to Legendre.
For a weak minimum it is necessary that
(3.3) fp.„. (x, yix), y'ix )) z.z, ^ 0 (t, j = 1, ■■■ , n),
for each point on g and every set ( z ).
One forms the function
<t>ie ) = E(x, yix), y'{x), y'ix) + ez),
where the fourth set of arguments is the set
y\(x) + ezi
(i — 1 > * y ■
[4]
THE JACOBI CONDITION
7
One readily finds that <£*(0) equals the left member of (3.3). From the Weier-
strass condition in the weak form it follows that has a relative minimum
when e — 0, and hence </>"( 0) ^ 0.
Condition (3.3) is thereby proved.
The Jacobi condition
4. We suppose again that g is a minimizing arc of class C1. We evaluate J
along the family of curves (1.4) obtaining thereby a function J(e). One readily
finds that the so-called second variation takes the form
J (0) - ( fliPjViVj +2 fliVViVi + f°yiViViVj) dx
where the superscript zero indicates evaluation along g. One sets
fliPjVWj + 2f°Piv/iVj + v^iVj = 2 to(r),v')
and
I(v) = J i 2 Q(rj, t?') dx.
For a minimizing arc we have J'(fi) = 0 so that it is necessary that 0) ^ 0.
Zero is then necessarily a minimum value of the second variation. To discover
the full consequences of this fact it is natural to consider the problem of minimiz¬
ing the second variation among admissible functions (77) (see Bliss [4]) and in
particular to consider the corresponding Euler equations
(4.1) ~ - a,. = 0 a - i, • • • , »>.
Equations (4.1) are termed the Jacobi equations. If formally expanded they
are linear and homogeneous in the variables rj % , 77 1- , rj { . The determinant of the
coefficients of rj { is | fp.p | evaluated along g. A point x = c at which this
determinant does not vanish will be termed a non-singular point of the Jacobi
equations.
In the neighborhood of a non-singular point x — a one can infer that a solution
of the Jacobi equations, if known to be of class C1, is necessarily of class C*,
that all solutions are linearly dependent on 2 n such solutions, and that a solution
(77) which with (77') vanishes at a, vanishes identically neighboring a.
We shall prove the following lemma.
Lemma. If there exists a solution (77) of the Jacobi equations which is of class
Cl on the interval a1 ^ x S a, a > a1, and which vanishes at the ends of the interval ,
then
0(77, 77') dx = 0.
(4.2)
8
THE FIXED END POINT PROBLEM
[I]
Because of the homogeneity of 9. in r/, and y \ we can write
/ 2 S2 dx = (Vi G„t. + Vi dx.
If the terms involving ^ be integrated by parts, we find that
/„ 2 0 = [* + /„ * |> - s dx-
The lemma follows at once.
Let (77) be a solution of the Jacobi equations of class C 2 on a closed interval of
the x axis bounded by distinct points x = c and x ~ a, and suppose that (77)
vanishes at the points x — c and x = a. If (rj) does not then vanish identically
between c and a neighboring x = a, the point a; = a on the x axis (or g) will be
termed conjugate to the point x = c on the x axis (or g).
The following theorem gives the Jacobi necessary condition.
Theorem 4.1. Jf g affords a weak minimum to J} no conjugate point of x = a1
on the interval a1 < x < a2 can coincide with a point at which the Jacobi equations
are non-singular .
A proof of this theorem has been given by Bliss [4] essentially as follows.
Suppose the theorem false, and that there exists a solution ( rj ) of the Jacobi
equations which is of class C2 on the interval a1 ^ x S a, a1 < a < a2, which
vanishes at x = ax and x — a without vanishing identically for x < a neighboring
x = a, where x = a is a point at which the Jacobi equations are non-singular.
Let 77* (r) be a set of functions equal to fji(x) on (a1, a), and zero on the remainder
of the interval (a1, a2). By virtue of the preceding lemma we see that I(v*) = 0.
But if g is a minimizing arc, as we are supposing, I(rj ) ^ 0 for all admissible
(77). Hence (77*) affords a minimum for I(rj).
It follows that (77*) must satisfy the Weierstrass-Erdmann corner conditions
at a; = a; that is on (77*) we must have
(4.3) = 0 (» = 1, • • • , »).
If we make use of the fact that (77*) vanishes at a, conditions (4.3) take the form
(4.4) /P^/(«) = 0 (i,j = 1 ,•••,»)
where the partial derivatives of / are evaluated at x = a on g. If the point a is
non-singular, we see from (4.4) that ( ff ') vanishes at a, and hence (ff) vanishes
identically near a. From this contradiction we infer the truth of the theorem.
Note. Strictly speaking the function 12 does not satisfy the requirements
imposed on /in §1, since it is merely of class C° in x. But even with Q of class
C° in x , one sees that the proof of the Weierstrass-Erdmann corner conditions
remains valid.
[5]
CONJUGATE POINTS
9
Conjugate points
5. It is necessary now to prepare for the sufficient conditions. To that end
we shall obtain £hree representations of the conjugate points of a point x — c.
For the remainder of this chapter we shall suppose that we have given an extremal
g along which
I fpiPj I ^ 0 (i, j = 1 , ■■■ , n).
Let || Vi,- ( x , c) || be an n-square matrix of functions of which the columns are
solutions of the Jacobi equations for c constant, and which satisfy the initial
conditions
(5-1) c ) = 0, vijx(c, c) = 81 (i,j = 1, • * , n),
where is the Kronecker delta. Such solutions exist according to the general
theory of differential equations. These n solutions are independent by virtue
of (5.1). Set
(5.2) D(r, c ) = | v„-(x, c) |.
The determinant D(x,c) vanishes at x — c in a way which we shall now determine.
We can set
Vij(xy c) = (x — c)aij(x) c)
(i, j
where
aaix, c) = jo lr +
t (x — c)> c] dt.
We see that ai}-(x, c) is continuous in x and c, and that
dij(c, C) == Viiz(c, c) = 8\.
We thereby obtain a representation of D(xy c) of the form
(5.3) D(x, c) = (a: — r)n A(xy c), A(cy c) = 1,
1, * • • , n)
where A ( x7 c ) is continuous in x and c, for x and c on an interval
(5.4) a1 — e < x < a2 + e, e > 0,
slightly larger than (a1, a2).
We note the following:
The conjugate points of a point x = c are the points x c at which D(xy c) = 0.
Let (v) be a proper linear combination of the columns of D(xy c), that is, a
linear combination in which the coefficients are not all null. If D(a, c) = 0
with a ^ cy there will exist a proper linear combination (v) of the columns of
D(xy c) .which vanishes at x = a. Moreover (v) is not identically zero near
x — c, and hence not identically zero near x = a. Thus x = a is a conjugate
point of x = c.
10
THE FIXED END POINT PROBLEM
[I]
Conversely if x = a is a conjugate point of x = c there must be a solution of
the Jacobi equations which is not identically zero, and which vanishes at
x = a and x = c. But all solutions which vanish at x = c are linear combinations
of the columns of D(x, c). Hence D(a, c. ) = 0, and the statement in italics is
proved .
Since D(x, a1) has an isolated zero at x = a1 there will either be no conjugate
point of x = a' on (a1, a2) or else there will exist a first such conjugate point
x = a > a1.
The following fact will be used later. If <*0 is the first conjugate point of c„,
and <*o and c0 are both on the interval (5.4), the first conjugate point, if it exists,
of a point r sufficiently near c0 will not precede a0 — e, where e is an arbitrarily
small positive constant. Upon referring to (5.3) we see for the case at hand that
(5.5) A(x, cfi / 0 (f, 5 i < a0).
Our statement follows at once from the continuity of A(x, c).
We now recall a principle discovered by Jacobi.
Let hi (x, n) be a one-parameter family of extremals which contains g for
M = Mo, and for which the functions hfix, p) are of class C- for M near Mo, and x on
(5.4). According to Jacobi the functions
(5.6) Vi(x) = fu fix, no) (t = 1, ■ • • , n)
afford a solution of the Jacobi equations determined by g.
To prove this fact we note that
d
fa'* hi?. i aO i hx(x, /x)] fy, \xy h(x, /i), hx(x, m)] = 0.
If we differentiate the left members of these identities with respect to /x, inter¬
change the order of differentiation with respect to x and /x, and set \l = /x0, the
resulting equations take the form of the Jacobi equations. One sees this the
more readily if one first verifies the fact that
~fPi = Ml, !»'), ~fUi = %(v,v'),
where (g) is given by (5.6) and the left members of these equations are evaluated
for m = Mo.
We turn next to the family of extremals y = yfix, x°, a, b) of §2. We suppose
that the extremal g is determined by the parameters (ar°, a, b ) = (c, a, 0). By
the above principle of Jacobi the functions
(5-7) ' Vi(x) = y^x, c, a, 0) (t, j = 1, • ■ • , n),
as well as the functions
(5.7) "
4»(^) 0) r
15]
CONJUGATE POINTS
11
afford solutions of the Jacobi equations for a fixed j. We here have 2 n solutions.
That these solutions are independent is readily proved. For upon suitably
differentiating the members of (2.5) we find that
ViaMf C> a>&) = ViaJC> c> a’ & = °»
(5.8)
Cy ay y ibjx^'i Cy ay &) ^ i •
We can now obtain another representation of the conjugate points of x = c
on (a1, a2). To that end consider the family of extremals
(5.9) y% = 4>i(x, b ) = y{(x, c, a, b) (t = 1, • • * , n),
passing through the point on g at which £ — c. The columns of the jacobian
(5.10)
, <j>n)
* * ' , 6n)
= #(*),
5. =
are solutions of the Jacobi equations. According to (5.8) they satisfy the same
initial conditions at x — c as do the corresponding columns of || t\,(z, c) ||.
'They are accordingly identical with these columns, so that D(x, c) becomes
identical with the jacobian (5.10). We therefore have the following lemma.
The conjugate points of x — c on the x axis are the zeros x ^ c of the jacobian
E(x).
A third representation of conjugate points is in terms of the so-called Mayer
determinant. With its aid we shall prove a lemma used in later chapters.
Together with the matrix |j t\-,(x, c) || previously considered, we introduce
here an n-square matrix || utJ{x, c) || whose columns represent solutions of the
Jacobi equations which satisfy the conditions
u,j(c, c) = S{, c) = 0 (i, j =!,•••,«).
We shall also consider a matrix || tup(x) || of n rows and 2 n columns, whose
columns represent a set of 2 n independent solutions of the Jacobi equations.
The 2n-square determinant
(5.11)
A(x, c )
V t p(e)
V<p(x)
(i = 1, • • ■ , n; p = 1, • , 2n)
is called the Mayer determinant. We shall determine its relation to the deter¬
minant D(xy c) previously considered.
We first verify the matrix identity
Vip(c)
Uij{c, c)
Viiic, c)
Vjp(c)
Vip(x)
Unix, c)
Vii(x, c)
Vjp (c)
This identity clearly holds if
(5. lo) tfipipj) — Uij{x, c]rijp{c) ^)7?7p(^) (u j !>’**> fty P ~ 1> ‘ ’ j 2n).
12
THE FIXED END POINT PROBLEM
[I]
But (5.13) holds for x = c, and the equations obtained by differentiating (5.13)
with respect to x hold for x = c. It follows that (5.13) holds identically. The
identity (5.12) then follows.
We next observe that
W(c)
Vipic)
ViP(c)
since the 2 n columns of \\ yip(x) || are independent,
thus obtain the important relation
From (5.12) we
(5.14) A(x, c) — D(Xj c)W(c), W(c) ^ 0.
The zeros of D(x, c ) are thus the zeros of A(x, c) = 0, so that we have a third
representation of the points conjugate to x — c.
We shall now prove the following theorem.
Theorem 5.1. Tf the point (xl, yl) on g is not conjugate to the point ( x 2, y2) on g ,
then any two points (xl, yl) and (x2, y2) sufficiently near (xl, yl) and (x2, y2) re¬
spectively, can be joined by a unique extremal which may be represented in the form
yi = x\ y\ x 2, y 2) (i = 1, ■ ■ • , n).
where the functions on the right are of class C2 in their arguments , for x on an in¬
terval slightly larger than the interval (xl, x2).
The family of extremals neighboring g has been represented in §2 in the form
yi = yfx, x°, a, b)
(i - 1, * • * , n).
Suppose that the set (x°, a, b) = (c, a , ft) determines g. To satisfy the theorem
we seek to solve the equations
(5.15)
y\ = c, a, b),
y\ = y<(x2, C, a, b),
for (a, b) as functions of (xl, yl, x2, y2). We have the initial solution
[x\ y1, X2, y2] = [x», y\ x2, y2], [a, 6] = [a, 0].
Moreover the jacobian of the right members of (5.15) with respect to the param¬
eters (a, b), evaluated at the initial solution, is readily seen to be the Mayer
determinant,
A(x2, x1) 5^ 0,
set up with the aid of the 2 n independent solutions (5.7). It is not zero since
x2 is not conjugate to x1.
We can accordingly solve equations (5.15) for (a, b) as functions
a.(z\ y\ x2, y2),
bi(x\ y\ X2, y2)
13
[ 6 ] THE HILBERT INTEGRAL
of the coordinates of the given end points. The functions
4>i(z, x\ y\ x 2, y2) = y^x, c , a(x\ y\ x2, y2)} b(z\ y\ z2, y2)]
will satisfy the requirements of the lemma.
The Hilbert integral
6. Let there be given an n-parameter family of extremals of the form
(6.1) Vi - Vi(x} ft, ■ • • , fin) (i = 1, • ■ • , n),
for which the functions 0) are of class C2 in the variables (r, fi) on some open
region R in the ( x , fi) space, and for which the jacobian
' ' ‘ > Vn) ^ ()
JKfih • • * , fin)
on R. If there is one and only one extremal of this family through each point
(xy y) of an open region S of the (r, y) space, the family of extremals will be
termed a field covering S. We suppose we have a field covering S.
The parameters (fi) corresponding to each point (x, y) of S will be functions
fii(Xj y) of f.r, ?/), of class C 2 on S. For (.r, y) on S we set
Pi(xy y) = ytr[j'y fi(Xy ?/)].
The functions pfixy y) are called the “slope functions” of the field. They
define the direction of the extremal through (x, y).
The Hilbert integral is a line integral of the form
1* = jA(x, y)dx + Bi(x, y)dyt ==/(/— Pifpi)dx +
where p, is to be set equal to pt(x, y). The expression used in the Hilbert integral
arises naturally enough, as we shall see in Oh. II, in the condition,
(/ ~ Pifn)dx + fpidyi = 0 (i = 1, • • • , n),
that the direction whose slopes are pfx , y) cut the direction whose slopes are
dyi/dx transversally at (xy y). Transversality is the natural generalization of
orthogonality.
As might be expected from this geometric setting, fields for which the Hilbert
integral is independent of the path joining two points in S have a peculiar
importance. They are called Mayer fields.
Let Rf be the part of the space (xy fi) that corresponds to S by virtue of (6.1).
The integral I* can equally well be represented by ^n integral on R'y with dx
and dfii as the independent differentials. Upon noting that
dy< ~ yixdx -f” yxphdfih (iy h 1, • • * , n)y
one obtains /* in the form
(6.2) 7* = fC(xy fi)dx + Di(Xy fi)dfii = ffdx + fPiy^dfihy
where yy and pi in / and fpiy are to be replaced by yi(xy fi) and 2/,*(z, fi )•
14
THE FIXED END POINT PROBLEM
[I]
If /* is independent of the path on S, it will be independent of the path on R',
and conversely. If R' is simply connected, the conditions that I* be independ¬
ent of the path joining two points take the form
— D, - ~ D,, = 0
d(ilt dfik
( h,k= 1, • • • , n).
Upon setting/^ = t\-(x, ft these conditions take the form
<6'3> aSW,-s("|) = 0'
<6'3>' k (" s) - k (’• S) - 0 (i- *■ 1 - ">•
Conditions (6.3) become
f d}ii -f „. ^ ^
aft ^ 1 aft, ax £ axaft a.r aft/
Upon making use of the fact that
f = ^ / _ a^t
conditions (6.3) reduce to identities.
Conditions (6.3)' are absent if n = 1. If n > 1, examples would show that
they are not in general fulfilled.
Although the left members of (6.3)' are not in general zero, we can prove that
(6.4)
- C,
where C is constant for each extremal, but may depend on (ft). To establish
(6.4) we evaluate the integral J along members of the field neighboring a partic¬
ular member of the field, between x = x0 and a variable x. We obtain thereby a
function J(xy ft. Upon differentiating J under the integral sign, and integrating
by parts in the usual way, we find that
Upon differentiating the right hand member of the first of these identities with
respect to ft and the second with respect to ft and equating the results, we find
that
r toi <ty± rv d2y i T s ay< d% T
Laft aft 4 aftaftj.. Laft, aft ^ 1 aftaftj */
[7]
This reduces to
SUFFICIENT CONDITIONS
15
dVi dy,
dih dy, x _
dVi dy,
dv, dyiT°
_dpk d/3>,
dffh
_dPk dpi, ~
dPi, dPk\ ’
and (6.4) is thereby proved.
Incidentally the proof of (6.4) does not depend at all upon the condition that
there be but one extremal through each point of S.
We can now readily prove the following.
The family of extremals passing through the paint x — c on g, if represented in the
form (5.9), forms a Mayer field, covering a neighborhood of any segment of g along
which the Jacobian of the family (5.9) does not vanish.
All the conditions for a Mayer field are clearly satisfied except possibly the
condition that the Hilbert integral.be independent of the path. But turning to
(6.4) we find that C = 0 for each extremal, as a consequence of the fact that
</>(c, b ) s constant. Thus the integrability conditions (6.3)' are satisfied, and
we have a Mayer field.
Sufficient conditions
7. The following theorem is due to Weierstrass, at least if n = 1.
Theorem 7.1. Suppose g is an extremal of a Mayer field which cotters a region
S including g in its interior. If
#0, y, p{x, y), q) > 0
for (:r, y) on S and any set Qi ^ p*(x, y)1 where pAx, y) is the ith slope function of
the field, then g affords a proper, strong minimum to J relative to all curves of class
Dl which join its end points on S.
Let 7 be an admissible curve yi = yA'x) joining the end points of g in S.
Corresponding to the given field we can set up the Hilbert, integral I* of §6.
Reference to (6.2) shows that J0 *-= 1*. But since 1* is independent of the path
joining its end points, /*=/*. Hence we can use the Hilbert integral to
represent J(, as follows :
(7.1)
v<fp) dx + /,>/%
where p, = />,(.r, y). Using our representation //,('.r) of 7, we have
J0
[ [f ~ PifPi + fP/i\dx
./a1
(* =!,••• , n),
where we understand that
Vi = 2 /<(*), y'i = y'i(x), Pi = Pi(x,y(x)).
16
THE FIXED END POINT PROBLEM
We are thus lead to the Weierstrass formula
(7.2) ./, - Ju = [ K[x, y(x), p(x, y(x)), y'(x)] dx.
Jr i»
Hence
(7.3) Jy - J0> 0,
unless K s 0 in (7.2), that is, unless ]ji(x) satisfies the differential equations
(7.4) ^ = Pdx, y) (t =1, ••• ,n).
But in such a case the uniqueness theorem of differential equation theory tells
us that 7 would coincide with the extremal of the field through its initial point,
that is, with g. "Thus (7.3) holds if 7 is different from g , and the theorem is
proved.
Before proceeding further it will be useful to enumerate certain conditions
that occur frequently hereafter. In all these conditions we suppose that we
have given an extremal <7, defined by y, = yfx), with x on the closed interval
(a1, a*).
By the Jacobi S- condition will be understood the condition that there be no
conjugate point of the initial point of g on g .
By the Legendre S-condition will be understood the condition that
fPiPi(x,y(r),y'(jr))ziZ, > 0
for x on (a1, a2) and (z) 9^ (0).
By the Weierstrass S-condition will be understood the condition that
E(x, y, p, q) > 0
for all sets (x, y, p) sufficiently near sets (x, y, y') on <7, and any set ( q ) 5^ (p).
The problem will be said to be positively regular in a region S of the (x, y)
space, if
fPiPj y, p) > 0
for (x, y) on Sr (p) unrestricted, and (2) any set not (0).
We come to the following theorem.
Theorem 7.2. In order that an extremal g afford a proper , strong minimum to J
relative to neighboring curves of class Dl which join its end points , it is sufficient
that the Weierstrass , Legendre , and Jacobi S-conditions hold relative to g.
A particular consequence of the Legendre S-condition, all that we use here, is
that | / | 0 along g. With this condition satisfied the results on conjugate
points in §5 apply. By virtue of the Jacobi S-condition we then know that the
first conjugate point of x ~ a1 on the x axis lies beyond x = a2 or fails to exist.
[7]
SUFFICIENT CONDITIONS
17
According to the results of §5, the first conjugate point of a point x - c prior to
x - a1, but sufficiently near x = a1, will lie beyond x - a2, or fail to exist.
Let § be the extremal obtained by extending g slightly. According to our
final result in §6, the family of extremals passing through the point x = c on Q,
if properly represented, will form a Mayer field in a sufficiently small neighbor¬
hood of g.
The present theorem follows from Theorem 7.1.
If the condition of positive regularity holds for ( x , y) near g, the Legendre
S-condition certainly holds for g. The Weierstrass N-condition also holds
relative to g. We see this upon using Taylor’s formula which shows that
E(x, y, p, q ) = - pd ( qj - pMnpfa y, p*) (*, i = !>•••,«)>
where
v*i = Pi + o(qi - p^, (o < e < l).
CHAPTER II
GENERAL END CONDITIONS
To the reader the objective of the present chapter may appear to be the
obtaining of necessary and sufficient conditions for a minimum under general end
conditions, and such conditions are an immediate objective. But in reality
steps are being taken towards a much larger goal.
Recall the analogy between function / and functional J, critical point and
extremal, quadratic form and second variation, the topology of the domain of
/ and the topology of the domain of J . We here take the first step towards
carrying out this analogy by assigning an index to an extremal, analogous to
the index of a quadratic form. This index is the number of negative character¬
istic roots in a boundary problem associated with the extremal.
There remain for later chapters most important problems. What is the
geometric significance of this index? Has it the property of invariance under
changes of coordinate system, or otherwise put, can it be given an invariantive
definition in a general parametric representation of the problem where over¬
lapping coordinate systems are used? What relation does this index bear to
other possible indices that could be assigned to the extremal in special cases?
In particular what relation does it bear to conjugate points, focal points, the
Poincar^ rotation number, the order of concavity of a periodic extremal, or to
other characteristic invariants of an extremal?
For the contemporary literature on the minimum problem under general end
conditions the reader is referred to the papers by Bliss, Carathdodory, Myers,
and to the Chicago Theses on the Calculus of Variations. The latter have-
appeared under the title Contributions to the Calculus of Variations , University
of Chicago Press. Further references will be found in these theses. The
preceding papers are primarily concerned with a minimum. The papers [8, 16]
of Morse and [1] of Currier are concerned not only with minimizing extremals
but also with the analytic and geometric characterization of extremals in general.
The end conditions
1. As in Chapter I we suppose that we have given an extremal g of the form
y{ = jji(: r), a1 g x g a2 (i = 1, • • • , n).
Points near the initial and final end points of g will be denoted respectively by
(*\ vl ••• ,y'n) = (*•, y*) (* * U 2)
where $ = 2 at the final end point and 1 at the initial end point.
18
[1]
THE END CONDITIONS
19
A curve of class Dl neighboring g will be termed admissible if its end points
are given by the functions
(1.1) x* = x*(ah • • • , *r)> y\ = y\(a ,, •••,«,) (0 g r g 2«. + 2),
for values of the parameters (a) near (0). For r = 0 the set (a) is vacuous, but
it will be convenient to understand symbolically that
x'(tt) = a4, y*(a) = &(«*).
For r > 0 and for (a) near (0) we suppose that the functions in (1.1) are of class
C2 and that they give the end points of g when (a) = (0).
For r > 0 let 0(a) be any function of (a) of class C 2. For r = 0, 0(a) shall
represent the symbol 0.
We seek the condit ions under which g and the set (a) = (0), vacuous if r = 0,
afford a minimum to the functional
r r-'l a)
(12) J = / fU, ?/, yf)dx + 0(a)
J a )
among sets (a) near (0), and admissible curves which join the end points x*(a),
In the classical treatment of the problem with general end conditions, Bolza
[2], Bliss [10], the end conditions have been given in the form of equations
(1.3) </>„(>, tf) = 0 (p = 0, 1, • • • , m ^ 2 n + 2)
with the restriction that the functional matrix of the functions <f>v with respect
to their arguments be of maximum rank. Bolza uses a function g(x% y*) in place
of our function 0(a). Conditions of the form (1.3) can be put in our form, but
not always conversely, and the function g(x\ if) can be reduced to our more
general 0(a) by means of (1.1). See Osgood [1], p. 155.
But the real reasons we have chosen to represent our end conditions in para¬
metric form are much deeper. The advantage of the parametric representation
of surfaces over a representation of the form <j>(xy y} z) — 0 has long been clear.
Corresponding advantages appear here when our end conditions are represented
in the form (1.1). In particular the algebraic problem of setting up the second
variation is much simpler and more symmetric. Moreover in the case where
(x1, yl) is required by (1.1) to rest on an w-manifold M while (r5, y 2) is fixed and
J is the arc length, the part of the second variation which appears outside the
integral sign is the second fundamental form of M except for a constant factor.
Our choice of end conditions in the form (1.1) was partly a matter of necessity.
We shall presently deal with end conditions given in the large. If one recalls
the fact that the only regular manifolds that can be represented by a single set
of parameters in a regular way are those with Euler-Poincar6 characteristics
zero, one sees that for the purposes of analysis, geometric configurations must in
general be represented by the aid of overlapping parametric systems. We must
20
GENERAL END CONDITIONS
[II]
not only use parametric representations, but must consider transformations
from one set of parameters to another.
An unexpected advantage of the form (1.1) was that it led to an algebraic
representation of the most general set of self-adjoint boundary conditions
associated with the Jacobi differential equations. As far as the author knows
this is the first representation of these conditions which contains just the con¬
stants which are arbitrary. New numerical invariants of these boundary
conditions thereby appear. In this way we are led to a natural and complete
class of generalizations of the Sturm-Liouville separation, comparison, and
oscillation theorems for the general self-adjoint system. (See Ch. IV.)
The transversality condition
2. Corresponding to the end conditions (1.1) our transversality condition is
written formally as follows:
(2.1) de + [(/ - Pif,,.) dx' + = o (t = 1, • • • , n),
where (x, y , p) is to be taken at the second end point of g when s = 2, and at the
first end point of g when s = 1. If r > 0, dd, dx8, and dy\ are to be expressed for
(a) = (0) in terms of the differentials dah, and (2.1) is to be understood as an
identity in these differentials. If r — 0, we have dd = dx8 ~ dy\ =0 so that
(2.1) is automatically satisfied. In this section we suppose that g is of class C1
and satisfies the Euler equations in the unexpanded form.
We shall prove the following theorem.
Theorem 2.1. A necessary condition that g afford a weak minimum to J rela¬
tive to neighboring admissible curves of class Cl is that it satisfy the transversality
condition (2.1).
The theorem is trivial in case r = 0. We suppose then that r > 0.
Points ( x •, y8) near the end points of g can be joined by a curve of class C1
neighboring g of the form
Vi = 9i(x) + [yi - Vi(x1)] + [(?/< - Vi (z2)) - (y\ - j/iOr1))] (*? •
We are here supposing that the functions y%(x) which define g have been extended
as functions of class Cl over an interval for x which includes the interval (a1, a2)
in its interior. If we set the variables x *, y$ respectively equal to the functions
z*(a), y\{a), we obtain a family of admissible curves y , = a) which join
the end points x*(a), y\{a), and reduce to g for (a) = (0). Let a*(e), h = h
• * • , r, be a set of functions of e of class C 1 for e near 0, with ah( 0) = 0. Set
<t> i(x, a(e)) = yi(x, e).
[3]
THE SECOND VARIATION
21
We have in yx — ?/»•(: r, e) a one-parameter family of admissible curves satis¬
fying the identity
(2.2) Ui(<x(e)) = yx[xH(a(e))y e].
We evaluate J along the curve of this family determined by the parameter e,
setting 0 = 0(a(e)), and taking the limits of the integral as
x* — x*(a(e)).
We thereby obtain a function J(e) such that
(2.3) J'(0) = fe + [/J + ^
If we differentiate the members of (2.2) with respect to e , we find that
(2.4)
dy\ , dx * , , ,
^7- = yiz(x% e) — + e).
We now eliminate yte from (2.3) by means of (2.4), and recall that ,/'( 0) = 0
for a minimizing arc g. We thus find that for e ~ 0
dO
de
_j_
(/ - f/irf,,)
dx *
de
+ f,
dy
' de
s "1 2
t _
- J i
0.
The transversality condition follows and the theorem is proved
The second variation
3. We have already obtained a formula for the second variation in the case
of fixed end points, that is, in the case r = 0. We now consider the case r > 0.
For r > 0, a set (a) in our end conditions (1.1) determines a pair of admissible
end points. A set of functions,
(3.1) ah ~ oifi(e), a/,(0) = 0,
will determine a set of such end points.
Suppose we have given such a set of functions «/,(<?) of class C2 for e near 0, and
a one-parameter family of curves,
(3.2) rji = y,[x, el
joining the end points determined by ah(e ) and reducing to g for e — 0. We
suppose that yx{xy e) is of class C2 for e near 0. We are also supposing that
(3.3) yi[x’(a(e)), e] = y\[a(e)) (i = 1, • • • , n; s = 1, 2).
For each value of e near 0 we evaluate J along the corresponding curve (3.2),
taking 6 as 0(a(e)) and taking the limits x 9 of the integral asx*(a(e)). We find
thereby that
(3.4)
22
GENERAL END CONDITIONS
[II]
We shall obtain a formula for the second variation ./"( 0). In it there appear
the variations of iji and an denoted by 7 u and uh respectively, and defined by the
equations
rji(x) =* yie(x, 0) uh = oc'h( 0) {i = 1, - • • , n)h = 1, • • • , r).
Before proceeding with the computation of J"(0) it will be convenient to
present two identities obtained by differentiating (3.3) with respect to e. Keep¬
ing the arguments as in (3.3) these identities are as follows:
(3.5)
(3.0)
. dx* dy
Vie + Viz Ik = IP
dx’ ( dx’ V
Vue + 2yiez Ik +yizx\7k)
+ Vi:
d?x* __ d2y\
' Id1 ” ~d?'
We return now to J'{e) and (3.4). Upon differentiating Jr(e ) with respect to
e and setting c ~ 0, we find that
./"(o) - ?! +
de 2 L de2
. d2x
f Z LT + M
/ d:rJ
\ de
)’
ris* (/?/* cix" dy^
+ Jvi de de de ~de
n 2
J 1
(3.7)
+
dx* I2 f"2
+ £ j"' + s,
where the superscript zero in the last term indicates evaluation for e = 0 prior
to carrying out the operation d/de. In carrying out this last operation we first
integrate by parts and then differentiate. This last term then reduces to the
following:
where the expression on the right is obtained with the aid of (3.6). W'e next
note that for x = x9(a («)),
(3.9)
dyix
de
“ 2/ixe r Z/ixx “T-
dx9
de 9
and that we can write (3.5) in the form
(3.10)
Viz
dx9
de *
We now make three replacements in (3.7). We replace the last general term
in (3.7) by the right member of (3.8) and replace the left members of (3.9) and
THE SECOND VARIATION
23
(3.10), where they occur in (3.7), by the corresponding right members of (3.9)
and (3.10) respectively. After these three replacements we find that
J'( 0) =
(3.11)
, , , \ d2x* f /dx8\ dx* dy* , . d2// ■
(/ Vixfp) + (/* Ihxfy) y~j~ J + ±fy.~ y| ”7“ + /;,
de dc
dc2
cPO f
+ ^+ / 2S2(W)*r,
a first form for the second variation.
But the terms outside the integral in (3.11) can be reduced to a quadratic form
in the variations uh = a*(0). To that end it will be convenient to denote
differentiation of x*y ?/*, and 6, with respect to ah or ak} by adding the subscript
h or k. At r — 0 we find that
(3.12)
h k 11 h 11 k "f" :V hah (9 b
^ //J A * M* W* +y’h"l( 0),
„ „ dx* dlj*. s ,
^h^k^h^k, -J = XhVxkUhUk,
^ = Ohkvhyk + eha*h( 0).
If the left members in (3.12) are replaced by the corresponding right members
in (3.12), we find that (3.11) takes the form
./"«)) = biluivl + Ota„m + [(/- // +/„,?/:*] J «I(0)
(3.13)
+
2Q(v, v')dx,
where we have set
hk = [(/- y',fPi)xlk + (L - y'ifv)xUl
(3-14) +fvUll>\k +xly\h) +f„ty’hh]] + ehk,
h, k = 1, • • • , r > 0; i = 1, • • • , n\ ft = 1, 2; .<? not summed.
These constants bhk are fundamental. We note that bi,k = (>/, /, •
The formula for J"(0) in (3.13) will be further simplified in case <j is a minimiz¬
ing arc, as we are assuming, by the fact that the coefficient of ah is null. This
follows from the transversality condition.
The variations rji and uh in (3.13) are not independent. In fact if we set
Tji(as) = 77*, (3.10) leads to the relations (« not summed)
nj = \y\h(0) -
24
GENERAL END CONDITIONS
[II]
For the purpose of reproducing these relations we write them in the form
v t " c { hUh (i lj * , n , h I, •** , r ^ 0) ,
(3.15)
c'.k = y‘ih( 0) - £<(a*)**(0) (s = 1, 2; s not summed).
We summarize as follows :
Theorem 3.1. If g is an extremal which satisfies the transversality conditions,
the second variation takes the form
(3.16) /"( 0) = hhkUkUk + J ' 2Q(rt, v')dx (h, k = 1, • • • ,r),
where (v) and the r constants ( u ) are respectively the variations yu{xt 0) anda'h (0)
{vacuous if r = 0), and satisfy the secondary end conditions ,
(3.17) rji — c\Kuh = 0 {i = 1, • • • , n) h = 1, • • • , r; s = 1, 2).
For r > 0 the constants bhk and c*h are given by (3.14) and (3.15) respectively.
For r = 0 they are not defined and disappear by convention from the preceding
relations. For r = 0 the secondary end conditions (3.17) take the form
Vi = 0 (f = 1, • • • , n; s = 1, 2).
Consider the case where J is the integral of the arc length and the end con¬
ditions require that the second end point be fixed while the first end point rests
on a regular n-dimensional manifold M of the form
xl = xl(ah • , an), y « = y\{oti, • * • , an).
One readily verifies the fact that the direction cosines of the tangent to g at its
initial end point A are
(3.18) f - ViSvi> f Pl) * ' * , f pn
for ( x, y} p) on g at A. The transversality conditions require that M cut g
orthogonally at the point A. Referring to (3.14) we see that
(3.19) bhkUkUk = [(/ - Pifpi)x lk + fPy)hk]uhuk.
Bearing in mind that the direction (3.18) is normal to Jkf at i, we see that the
right member of (3.19) gives the terms of second order in the distance from the
point (a) = ( u ) on M to the n-plane tangent to M at A, except for a factor ±1/2.
Thus the form (3.19) is a second fundamental form of M at A. The implica¬
tions of this fact both here and later could be pursued to advantage much
further, but lack of space prevents such developments.
The accessory boundary problem
4. One can assign an index to a given quadratic form
(4.0)
aijZiZj
(i, j = 1, • • , p)
[4]
THE ACCESSORY BOUNDARY PROBLEM
25
in the following way. Setting up the characteristic form
(4.0)' Q(z) = dijZiZj - \ZiZif
recall that the necessary conditions that Q have a minimum include the con¬
ditions
(4.0)" Qn(z) = 2 (dijZi - \Zi) =0 (i = 1, • • • , p).
Numbers X* which with the sets (z) ^ (0) satisfy (4.0)" are called characteristic
roots. The index of the form (4.0) can be defined as the number of negative
characteristic roots (counted suitably if multiple).
Each of the above steps has its analogue in the theory of the second variation.
The analogue of the form (4.0) is the functional
bhkUhUk + J 20 (rj, i}')dx (A, A* = 1, • * • , r)
subject to the secondary end conditions
(4.1) Vi - c*ihuh = 0,
ivhile the analogue of the characteristic form (4.0)' is the functional
(4.2) I(i 7, X) = bhkUhuk + J ^ (20(t?, ?/') — Xvtlddx,
again subject to (4.1).
A set of n functions vi(%) and r constants (u) will be termed admissible if
rn(x) is of class Dx on ( a !, a2) and if (v) with the r constants (u) satisfies the second¬
ary end conditions (4.1).
The analogue of the conditions (4.0)" is a set of necessary conditions that an
admissible (rj) with r constants (u) afford a minimum to I( 77, X) relative to
admissible sets (rj) and ( u ), with X fixed. These necessary conditions include
the differential equations
~ fl,'. — + Xi/t = 0 (i = 1, • • * , n),
dx 1 1
and the transversality condition requiring that the condition
(4.3) [2fy' dv^ + = 0 (r > 0)
be an identity in the differentials duh. Upon using (4.1), (4.3) reduces to the
conditions
+ btjcUk = 0 (A, fc = 1, • • * , r).
If we set
ft(x) = r/'(x)], f J = fi(a*),
2G
GENERAL END CONDITIONS
[HI
(4.3J reduces to the conditions
(4.4) ~~ clih'([ + bhkUk = 0 (i = 1, ■ , n; A, t = 1, • • • , r > 0).
JFe term conditions (4.4) Me secondary transversality conditions.
The analogue of the conditions (4.0)" Ls the set of conditions
(4 .ay -f o,; - £4. + xt,,. = o (i = i, • • • , »),
(4.5) " 17* - «:*M* = 0,
(4.5) "' - ci*fl + = 0 (h, k = 1, • ■ • , r),
defining what we call the accessory boundary problem associated with the extremal g.
For r = 0 the conditions (4.5)"' disappear under our conventions, and the
conditions (4.5)" reduce to 77* = 0. For r = 0 we understand that the set (u)
is empty.
We note that the boundary conditions in t he accessory boundary problem are
composed of the secondary end conditions and the secondary transversality
conditions.
By a solution of the accessory boundary problem is meant a set of functions
7 n(x) which are of class C2 on (a1, a2) and which with a constant X, and r constants
(u), satisfy the conditions (4.5) If (77) # (0), the solution is called a character¬
istic solution and X a characteristic root. By the index of a characteristic root X
is meant the number of linearly independent characteristic solutions (77) cor¬
responding to the root X.
The final analysis of the index of the second variation subject to the secondary
end conditions will be deferred to the next chapter. Under conditions of
regularity to be given presently this index will be defined to be the number of
negative characteristic roots, each counted a number of times equal to its index
The necessary condition on the characteristic roots
5. We shall now prove the following theorem. Its analogue in the theory of
quadratic forms or of functions of several variables is clear.
Theorem 5.1. If an extremal g affords a weak minimum to J relative to neighbor¬
ing admissible curves, there can exist no characteristic root X < 0.
In proving this theorem we shall make no assumption concerning the value of
| | along (j.
We begin with the following lemma, r ^ 0.
Lemma. If (77) is a characteristic solution satisfying (4.5) with r constants (u)
and root X, then for these constants 7(77, X) = 0.
To prove the lemma we write 7 in the form
7(77, X) = bhkUhuk + / (77*12, + — \yem)dx,
Ja > *
15]
NECESSARY CONDITION ON CHARACTERISTIC ROOTS
27
and integrate the terms involving rj • by parts, in the usual way. We find for
the given (77), X, and r constants (w), that
(5.1) /(??, X) — bhkUhih +
If r = 0, the set (u) is empty, 77 ■ = 0, and the lemma follows from (5.1). If
7 > 0, we multiply (4.5)"' by uh, sum with respect to A, and use (4.5)". We
thereby find that the right member of (5.1) is null.' The lemma is thereby
proved. We return to the theorem.
Corresponding to the given characteristic solution (77) satisfying (4.5) with r
constants (u), we shall exhibit an admissible family of curves yt = yl(x, e), which
is of the nature of the family (3.2), which reduces to g for e = 0, which satisfies
the original end conditions with r parameters, (a) = (eu), and whose variations
are 77 ,(x). More precisely, y{(x , c) shall satisfy the identities
(5.2) V\(eu) = Vx{xa(eu), c] (s = 1, 2),
(5.3) ?/i Ax, 0) ^ t)i(x).
For r > 0 such a family is given as follows:
Vi(x, e) = yt(x) + erjt(x) - \y%[x?(eu) ) - y\{eu) + er)t[x2(eu)] | -y— - ■
x\eu) — x\eu)
(5.4)
- [ftfr'Ml - »; M + -.[*■(«.)] 1
That this family reduces to g for e = 0 and satisfies (5.2) is verified by direct
substitution. To verify (5.3), it is convenient first to observe that the brace
B = \yt[x*(eu)] - y\{cu) + erjt[x*(en)]\
is zero for e = 0. Moreover at e — 0
dB
de
{(y - y*ih(fi))uh + v9i I = 0,
as follows from (3.15). The identity (5.3) is now verified with ease.
For r = Owe set yt(x , e) — yi(x) + ern(x).
For the family ?/t(x, e ) so defined we know that
J"( 0) = bhkUhUk + j 211(17, 77 r)dx.
But by virtue of the lemma this becomes
J"( 0) = X J ViVidx
where X is the characteristic root associated with r)i(x). If X < 0, «/"( 0) < 0.
But this is impossible if g is a minimizing arc. Hence there can be no negative
characteristic root and the theorem is proved.
28
GENERAL END CONDITIONS
[II]
The non-tangency hypothesis
6. Before proceeding to the sufficient conditions it is convenient to introduce
an hypothesis which distinguishes a general case from a special case.
In ordinary problems involving transversality of a manifold to a given ex¬
tremal it is generally customary to assume that the manifold is not tangent to
the given extremal, or to insure this by other assumptions. There is here a
corresponding assumption. In the case where the assumption is not made
sufficient conditions involving the characteristic roots have been obtained by the
author, Morse [8], and Myers [3], but the results are much simpler in case the
assumption is made. Moreover simple examples in the plane will show the
relative unimportance of the special case.
In the space of the 2n+2 variables ( x *, ya) consider the 2-dimensional mani¬
fold defined by the equations
(6.1) y\ = Hx*) (s « l,2;t = 1, ••• , n).
This manifold is essentially the arbitrary combination of a point of g near the
final end of g with a point of g near t he initial end of g. We call it the extremal
manifold . The manifold
y'i = X ’ = X*(or)
in the same (2n+2)-space will be called the terminal manifold.
In case r > 0 we shall assume hereafter that the terminal manifold, is regular ,
that is, that the functional matrix
xm
(6.2) (s = 1, 2; h = 1, • • * , r; z = 1, • • • , n)
Vih(fl)
is of rank r.
If r > 0, our non-tangency condition is the condition that the extremal manifold
and the terminal manifold possess no common tangent line at the point («) - (0).
If r = 0, we make the convention that the non-tangency condition is fulfilled.
We shall prove the following lemma.
Lemma 6.1. In case r > 0 a necessary and sufficient condition for the non-
tangency condition to hold is that the matrix 1 1 c *ih 1 1 of (3.15) be of rank r.
A set of direction numbers of the tangents to the parametric curves on the
terminal manifold at (a) •= (0) is given by the r columns of the matrix (6.2).
At the same point direction numbers of the tangents to the parametric curves
on the extremal manifold are given by the two columns of the matrix
1 0
0 1
Si (a1) 0
0 y’^a?)
(6.3)
[6]
THE NON-TANGENCY HYPOTHESIS
29
The non-tangency condition implies that there is no linear relation between the
columns of the matrices (6.2) and (6.3) which actually involves both matrices.
Suppose the non-tangency condition failed. There would then exist con¬
stants ah not all zero, and constants k9 not both zero, such that
(6.4) ' ahx*h = k9 (h = 1, * • • , r; s = 1, 2),
(6.4) " a,ky*ih — kay'i(a9) (i = 1, • • • , n; s not summed).
Upon eliminating k 9 from these two sets of relations we find that
(6.5) ah[y*ih - x^y^a9)] = ahc\h = 0,
so that \\c\h || could not be of rank r.
Conversely, suppose || c\h || were of rank less than r. Then relations (6.5)
would hold with constants ah not all zero. If, moreover, constants k9 are defined
by the equations
(6.6) ahxeh = k9,
the relations (6.5) take the form (6.4)". We thus have a relation between the
columns of the two matrices (6.2) and (6.3) actually involving both matrices
unless both constants k9 = 0. But both constants k 8 can not be null because it
would then folldw from (6.6) and (6.4)" that the matrix (6.2) would not be of
rank r, contrary to hypothesis.
Thus if j | c || were of rank less than r, the non-tangency condition would
fail.
The lemma is thereby proved.
The variations (77) and r constants (u) appearing in the second variation
satisfy the relations
(6.7) ?? • - c\huh = 0,
as we have seen. If r > 0 and the non-tangency condition holds, we can solve
(6.7) for the variables ( u ) in terms of a suitable subset of r of the variations
77*. We thus have the following lemma.
Lemma 6.2. If the non-tangency condition holds} the second variation can he
written in the form
(6.8) J"( 0) = q(v) + 20(17, v')dx,
where q(jj) is a quadratic form in a suitable subset of r of the variations 77 J.
In case r = 0 the form q(r}) disappears.
Another advantage of assuming that the non-tangency condition holds is that
the accessory boundary conditions
(6.9) 77^ c ihUk ^ 0 ihy k — 1, ••• , r, i — 1, } n, s 1, 2),
(6.10) t\c\h - r \c\k + bhkuk = 0
30
GENERAL END CONDITIONS
[II]
can then be reduced to 2 n linearly independent conditions
(6.11) L„(v, f) = 0 (p = 1, ••■,2n)
on the variables y J , f J .
For we can eliminate the variables (u) from (6.9), leaving 2n — rndependent
linear conditions on the variables y ". Upon replacing the variables (u) in (6.10)
by linear combinations of the variables y*t obtained from (6.9), we obtain r more
conditions on the variables y\y f J, independent of the 2n — r conditions on the
variables y*{ already obtained from (6.9). The 2 n independent conditions (6.11)
thus result. If r = 0, they are the conditions y\ — 0.
The form Q(u , X)
7. For the remainder of this chapter we shall assume that g is an extremal
satisfying the transversality and non-tangency conditions as well as the Legendre
nS-condition. As previously we suppose that g is given in the form yx = yt(x).
We here introduce the functional
/x = 8(ot) + [ [f(x, y, y') (Vi ~ yi(x))2)dx
subject to the end conditions (1.1). For each X, g will still be an extremal and
satisfy the transversality and Legendre ^-conditions relative to /x.
The Jacobi equations are now the differential equations
(7.0) ~ ^ ~ + \y i = 0 (i = 1, • • • , w)»
where il is defined as previously. For each X conjugate points are to be defined
as previously in terms of solutions of (7.0).
We shall now define a quadratic form Q(u, X). We shall define Q only for the
case in which r > 0 in the end conditions.
For r > 0 let ?/»(: r, e) be any admissible family of curves of the same nature as
the family (3.2), satisfying the end conditions for ah == och(e)y and reducing to
g for e — 0. The second variation here takes the form
cP n f a*
(7.1) ------ = bhkUhUk + I [2f2(tf, yf) — \yiyi]dx = I(y , X)
where r n and uh are respectively the variations yie(x, 0) and a^(0), and the
constants bflk are defined as before. As previously we have
(7.1) ' y\ - c*ihuh (i = 1, • • • , n; h = 1, • • • , r; s = 1, 2).
If for a given X the end points of g are not conjugate, then the end points
x*{a)> y*i(a) can be joined for each (a) sufficiently near (0), by an extremal of
the form
Vi = a), 0) =
[7]
THE FORM Q(u, X)
31
where, as we have seen in Theorem 5.1, Ch. I, the functions <*>,•(*, <*) are of class
C2 in x and (a). We then let J(a, X) denote the value of JK taken along the
extremal determined by (a).
All of the first partial derivatives of J(a, X) with respect to the variables (a)
vanish when (a) = (0). This appears as an easy consequence of the trans-
versality conditions.
The terms of the second order in J(a, X), for A constant, now come to the
fore. They will be obtained by means of the relation
(7.2)
Jahak( 0, X)llhUk
SJ(eU"
eur, X)
where (u) is a set of r constants, and e is a parameter neighboring e = 0. To this
end consider the family of extremals
(7.3)
y% = yi{x, e) = eu), ah = euh.
The right member of (7.2) is simply the second variation of Jx for this family
so that
(7.4) JraA«ik( 0, \)uhuk = bhkuhUk + J [20(77, 77') - XrjtTji]dx
where the set (u) is the set ( u ) used in (7.3), and rjt(x) — y^(xf 0).
By the Jacobi principle the functions (77) appearing in (7.4) are solutions of
(7.0) for the given X, defining what it is convenient to call a secondary extremal.
This secondary extremal satisfies the secondary end conditions (7.1) 7 and is
accordingly completely determined by the constants (u).
We summarize these results as follows:
Suppose the end points of g are not conjugate for a given X. Let the value of Jx,
taken along the unique extremal joining the end points determined by («) for (a)
near (0), be denoted by J(a, X). Upon setting
Q(u, X) = Jahak( 0, \)uhuk (h, k = 1, • • , r > 0),
we find that
(7.5) Q(u, X) = bhkUhUk + j [20(77, 77') — Xt?^ i]dx}
where
(7.6) 77^ - c\huh - 0 (i = 1, * • • , n; h = 1, • • • , r; s = 1, 2)
and (77) is on the unique secondary extremal joining the end points (x, 77) = (a*, 77").
We shall say that I( 77, X) is positive definite for a given X subject to (7.6), if it is
positive for all curves (77) which are of class Dl, which satisfy (7.6), and on which
<v) & (0).
We shall prove the following theorem.
32
GENERAL END CONDITIONS
[II]
Theorem 7.1. If the Legendre S-condition and the non-tangency condition hold,
then I{t], A) is positive definite subject to (7.6), for — A sufficiently large, and r ^ 0.
According to Lemma 6.2, subject to (7.6) we have
(7.7) l(v, A) = q(v) + £ (29. - \ViVi)dx
where qfa) is a quadratic form in the variables r)\.
Now any such form as qfa) will satisfy a relation
(7.8) qfa) ^ -c[r}]v) + vWi ] (t = 1, • • • , n)
provided c be a sufficiently large positive constant . If h(x) is any function of x
of class Cl on (a1, a2), such that
h(al ) - -1, h(a 2) = 1,
then (7.8) can be written in the form
(7.9) q(v) Z-cJ°~ lh(x)ViVi)dx,
where ( rj ) represents any set of functions of x of class I)1 on (a1, a2).
From (7.9) we see that
r a* J
Hv, X) ^ / [212 — \77i77i ~ C — (/?T7i77i)]<^X.
7a > ^
Under the integral sign we have a symmetric quadratic form
(7.10) H{nu ,v'n>v i, *■*
We shall use the Kronecker rule for determining the index of the form H. To
that end we set A0 = 1, and Ak equal to the determinant of the form obtained
by setting the last 2n — k of the variables equal to zero in H .
According to Kronecker the index of the form H is the number of changes of
sign in the sequence A 0, Alf • • • ; A2ny if the form is regularly arranged. See
Dickson [1], p. 81. If one notes that the terms in H due to the introduction of
<7(7?) under the integral sign do not involve any terms quadratic in rjfi9 one sees
as a consequence of the Legendre S-condition, that the numbers A0, A\y • • • ,
An are all positive. Moreover the remaining AVs all become positive for —X
sufficiently large. Hence for —X sufficiently large, the form H is positive
definite. Hence I fa , X) is positive definite subject to (7.6) and the theorem is
proved.
We shall verify the following corollary.
Corollary. For r > 0 in the end conditions and for — X sufficiently large , the
form Q(u, X) exists and is positive definite.
[8]
SUFFICIENT CONDITIONS
33
For —X sufficiently large the integral
£ mv
, y') ~ A? uvi\dx
is positive except when (77) s (0). It follows that the end points of g cannot
then be conjugate. Hence the construction by which the function J(a, A) was
set up is valid. The form Q(uy A) then exists, and by virtue of the preceding
theorem and (7.5) it must be positive definite for - A sufficiently large.
Sufficient conditions
8. We continue with the hypotheses made in the first paragraph of the last
section.
We shall prove the following lemma.
Lemma 8.1. If for a given A there are no pairs of conjugate points on g} if the
Legendre S-condition holds , and in case r > 0 if Q(uf A) is positive definite , then
subject to the secondary end conditions
(8.0) v’i ~ c’ihuh = 0 (h - 1, • • • , r £ 0),
the functional I (77, A) is positive definite.
In the problem of minimizing /(rj, A) certain facts should be observed.
The corresponding Jacobi equations will be the same regardless of what
particular secondary extremal y is regarded as the extremal g. For a fixed A
the distribution of conjugate points on (a1, a2) will then be independent of 7.
Further a field of secondary extremals which covers a segment (a, b) of the x axis
can be extended so as to cover a region in the (xf 77) space which includes all
points between the n-planes x — a and x = b, as follows from the fact that the
coordinates 77* on extremals of such a field can all be multiplied by an arbitrary
constant k 0, and still form a field of extremals.
We also note that the Legendre ^-condition for g is the condition of positive
regularity for I(rjf A) for all points (x, 77) between the planes x = a1 and x = a7
inclusive, thus giving the strongest sufficient condition in its category for /.
To turn to the lemma we see that each secondary extremal defined on (a1, a2)
gives a proper minimum relative to admissible curves which join its end points.
The lemma is accordingly true if r = 0.
If r > 0, the end points of y>(x) are determined by constants ( u ) in (8.0), and
we have
I(y, A) ^ Q(u, A),
the equality sign holding only when (? 7) is a secondary extremal. Since Q(u , A)
is positive definite, I is positive except when (77) == (0). Thus I is positive sub¬
ject to (8.0), provided (77) ^ (0).
The lemma is thereby proved.
We shall now prove the following lemma, r ^0.
34
GENERAL END CONDITIONS
[II]
Lemma 8.2. If all characteristic roots are positive , then for A ^ 0 and sets (77)
subject to the secondary etui conditions , /(?;, A) fs positive definite.
We are assuming that the trarisversality and non-tangency conditions hold for
g , as well as the Legendre >S-condition. If the lemma were false, there would
be a least upper bound A0 ^ 0 of the values of A for which 7(r 7, A) is positive
definite. We would then have
(8.1) I(rj, A0) ^ 0
for all admissible curves (77) satisfying (8.0).
I say that the equality must be excluded in (8.1) for any admissible curve
( f) ) (0), satisfying (8.0). For if 7(fj, A0) = 0, (fj) would afford a minimum to
I(rj) A0) among admissible curves satisfying (8.0). It would follow that (ff)
could have no corners, would satisfy the secondary transversality conditions,
and hence be a characteristic solution. Hence A0 would be a characteristic root
contrary to our hypothesis that there is no characteristic root A0 ^ 0.
We conclude that I(rj, A0) > 0 unless (77) = (0).
Thus the segment (a1, a2) of t he x axis affords a proper minimum to 7 (77, A0)
relative to admissible curves (77) satisfying (8.0). According to the Jacobi
necessary condition there can be no conjugate points of x = a1 on the open
interval (a1, a2), for A -- Au. Since /( 77, A0) is positive definite, x ~ a2 cannot
be conjugate to x = a1.
Not only will there be no conjugate point of x = a1 on (a1, a2) for A = A0, but
it also follows from our representations of conjugate points in §5, Ch. I, that
there will be no conjugate point of x = a1 on (a1, a2), for A slightly in excess of
A0. Hence, in case r = 0, A0 cannot be the least upper bound of the values of A
for which 7( 77, A) is positive definite.
We turn now to the case r > 0. Since x — a1 is not conjugate to x = a2 for
A = An, and hence for A sufficiently near A0, our construction of Q(u , A) is valid
for A sufficiently near A(). From (7.5) we next see that Q(u, A0) must be positive
definite. It will accordingly be positive definite for values of A slight ly in excess
of A0. We can then infer from Lemma 8.1 that l(rjy A) is positive definite for A
slightly in excess of A0. But this is contrary to the choice of Ao. The lemma is
thereby established in case r > 0.
The proof is complete.
We now state the basic sufficiency theorem of this chapter, r ^ 0.
Theorem 8.1. In order that an extremal g afford a proper , strong , relative
minimum in our problem it is sufficient that it satisfy the transversality and non-
tangency conditions , the Legendre and Weierstrass S-conditionsy and that all char¬
acteristic roots be positive.
We first consider the case r > 0.
It follows from Lemma 8.2 that the end points of (a1, a2) ai j never conjugate
for A S 0. Accordingly the form Q{uy A) exists for each A ^ 0, and turning to
Lemma 8.2 again, we see that Q(u, A) is positive definite for each such A.
Let J (a, 0) be represented by means of Taylor's formula with the remainder
[8]
SUFFICIENT CONDITIONS
35
as a term of the second order in the variables (a). This remainder is approxi¬
mated by the positive definite form Q(a , 0)/2 in such a fashion that we can be
assured that
(8.2) J(a, 0) > J( 0, 0)
for all sets (a) ^ (0) sufficiently near (0).
Now if the Legendre ^-condition holds along g and there are no pairs of con¬
jugate points on g, one sees readily from the form of these conditions that they
also hold when g is replaced by a neighboring extremal segment ga with end points
determined by a set (a) sufficiently near (0). Moreover the field of extremals
which was used to prove that g afforded a minimum to J in the fixed end point
problem can now be similarly defined for each extremal ga . To that end we take
a family Fa of extremals issuing from the point on ga at which x = a1 — e}
where e is a small positive constant. For e sufficiently small, for (a) sufficiently
near (0), and for a set of initial slopes (p) sufficiently near those on ga at x
=■ a1 — e, it is seen that each family Fa will form a field covering a neighbor¬
hood N of g independent, of Fa.
Each extremal ga for which (a) is sufficiently near zero will then afford a
minimum to J relative to admissible curves 7 which join its end points and lie
on a sufficiently small neighborhood A7' CZ N of g. Thus on Nf
(8.3) J7 J(a, 0),
the equality holding only when 7 — ga. From (8.2) and (8.3) we see (hat
Jy ^ J( 0, 0),
the equality holding only if 7 is an extremal ga and ga is the extremal g.
The theorem is thereby proved in case r > 0.
In case r = 0 we see from Lemma 8.2 that I(rj, 0) is positive definite. It
follows that there is no conjugate point of x = a1 on (a\ a?) for X ~ 0. The
theorem then follows from the sufficiency theorem of Oh. I.
Since the Legendre ^-condition entails the Weierstrass ^-condition in its
weak form, we have the following corollary of the theorem.
Corollary. The conditions of the theorem , omitting the Weierstrass S -condition
are sufficient for g to afford a proper , weak, relative minimum to J
We also note the follow ing :
In the theorem, the condition that all characteristic roots he positive can he replaced
by the condition that the second variation he positive definite subject to the secondary
end conditions.
The proof, in Lemma 8.2, that 7(77, 0) is positive definite subject to (8.0), leads
with obvious changes to a proof of the following theorem.
Theorem 8.2. If g satisfies the transversality and, non-tangcncy conditions , if
the Legendre S-condition holds , and if \ = 0 is the smallest characteristic root , then
the second variation will he positive for all admissible sets (rj) ^ (0) except for those
characteristic solutions for which X = 0.
36
GENERAL END CONDITIONS
[II]
Our theorems take a particularly simple form for the case of periodic extremals.
Here we suppose that the integrand / and the extremal g have a period o> in x
and that a1 — a1 — o>. We compare g with the neighboring curves of class Dl
whose end points are congruent , i.e., whose y-coordinates at x — a1 and x — a2
are the same. We can take these common ^-coordinates as the end parameters
(a). Thus the end conditions take the form
y\ = «»> x* = a* (i = 1, * • • , n; * = 1, 2).
We also suppose that 6(a) s 0.
From (3.12) we see that bhk = 0. The accessory boundary conditions take the
form
— fo (i = 1, • • • , n).
Any non-null periodic solution of the accessory boundary problem is then a
characteristic solution.
The transversality conditions are automatically fulfilled, as well as the non-
tangency condition.
We have the following theorems.
Theorem 8.3. In order that a periodic extremal g afford a weak minimum to J
relative to neighboring curves of class Dl joining congruent points , it is necessary
that the accessory differential equations for X < 0 have no periodic solutions
to * (0).
Theorem 8.4. In order that a periodic extremal g afford a proper , strong mini¬
mum to J relative to neighboring curves of class Dl joining congruent points , it is
sufficient that the Legendre and Weierstrass ^-conditions be satisfied , and that the
accessory differential equations for X rg 0 have no periodic solutions ( rj ) ^ (0) .
The importance of the study of the relations between the calculus of variations
and the theory of characteristic roots in the associated linear boundary problems
has been revealed in many significant ways by Hilbert and Courant [1] in their
well known treatise on mathematical physics. In (n + l)-space with the
general end conditions in non-parametrie form, Cope [1] first obtained the
necessary condition on the characteristic roots. See also Bliss [10]. Sufficiency
conditions involving characteristic roots in the general problem in the Bolza
form of the Lagrange problem were first established by the author, Morse [8,
14, 15, 16]. The sufficiency conditions of the present chapter and their proof can
also be readily adapted to the Lagrange problem for the case of an extremal that
is identically normal.
In this connection the author has recently obtained what is believed to be the
first proof of the following theorem. An extremal in the Lagrange problem for
which the first multiplier can be taken as unity will afford a minimum in the
fixed end point problem, provided the usual Jacobi, Legendre and Weierstrass
sufficient conditions hold. The hypotheses of this theorem admit cases where
the family of extremals through a point fail in general to form a field. The
proof will be published in the Transactions of the American Mathematical
Society.
CHAPTER III
THE INDEX FORM
In this chapter we shall deal with the functional
J = 0(a) + f(x, y, y')dx,
subject to the end conditions
(0.1) x‘ = x'(a), y\ = y * (a) (i = 1, • * • , n\ s = 1, 2),
as described in §1 of the last chapter. Except for a temporary diversion in
§4, where we shall establish a necessary condition, we shall assume that we have
an extremal g satisfying the transversality and non-tangency conditions, as well
as the Legendre S-condition.
We define the index v of such an extremal to be the number of negative char¬
acteristic roots X in the accessory boundary problem, counting each root a number of
times equal to its index.
Such an index of g may also be regarded as the index of the quadratic func¬
tional given by the second variation subject to the secondary end conditions
(0.2) - c\huh = 0 (i = 1, • • • , n; h = 1, • • • , r; s = 1, 2)
of the preceding chapter, thus generalizing the notion of the index of a quadratic
form. We shall show that this index is finite. With the extremal g we shall
associate an ordinary quadratic form Q to be called the index form. The index
of Q will turn out to be the index of g. This index form is the key to all sub¬
sequent analysis in the small.
We use the index form to treat the problems with one or two end manifolds
and the problem with periodic end conditions. New invariants are introduced
and results of generality and refinement are obtained. See Morse [3, 5, 7, 10,
16, 17].
Definition of the index form
1. We begin with the following lemma.
Lemma. A decrease of X never causes a decrease of the distance from a point x = c
to the first following conjugate point.
Suppose x = Ci is the first conjugate point following x = c for X = Xi. Let
c0 be a value of x such that
C < C o < Cl.
37
38 THE INDEX FORM [ III ]
According to the sufficient conditions in the fixed end point theory, the integral
[212(77, ft) — Xi 7]i7ji]dx
will be positive on all curves ( rj ) ^ (0), of class D1, vanishing at c and c0. If now
Xi be replaced by a smaller constant X0, the integral will be positive as before.
Hence no point c0 between c and cy can be conjugate to x — c for X = X0, and the
lemma is proved.
For any fixed value of X there will exist a positive lower bound d(\) of the
distances between pairs of conjugate points on (a1, a2). This follows from the
representation of conjugate points by means of the zeros of the function A (x, c)
of (5.3), Ch. I. By virtue of the preceding lemma a lower bound d(X0) will
serve as a similar lower bound for all smaller values of X.
With this understood let
(1.1) X = a0, • • • , X = a,p + 1 (flo = a1; ap f i = a2)
be a set of points on the x axis, arranged in the order of their subscripts, and
chosen so as to divide (a1, a 2) into segments of lengths less than d(X). Let us
cut across g at the point at which x = a g by an 7/,-dimensional manifold M„
of the form
(1.2) x = A'"(/3), yi = V?(/3l (ry = 1, • • • , p),
where the functions Ar"(ft and FJ( ft are of class C 2 in the parameters
(/3) = (ft, ft, • • ■ , ft)
for (ft near (0), We suppose that Mq intersects g at x = aq when (/3) = (0),
but is not tangent to g. We suppose further that the representation of M q at the
point (0) — (0) is regular ; that is, not all of the jacobians of n of the functions
Xq, F® with respect to the n parameters (ft are zero at (ft = (0). We term
the manifolds M q the intermediate manifolds.
Let
(1.3) Ph • • • , Pp
be a set of points neighboring g on the respective intermediate manifolds Mq.
Let (v) be a set of pn + r = 5 variables of which the first r shall equal the r
parameters (a). The next n (the first n if r = 0) shall equal the parameters
(ft of the point P\ on M h the next n those of the point P% on M2, the next a
similar set for P3, and so on to Pp. If r > 0, a set (a) determines the end points
&(«), yl(a)) = Po,
02(a), ?/2(a)) = Pp + 1.
The complete set ( v ) determines the points
(1.4) Po, * * * , Pp + 1,
[1]
DEFINITION OF THE INDEX FORM
39
and for points (1.4) sufficiently near g is uniquely determined by these points
(1.4) . For ( v ) — (0) these points lie on g .
If the points (1.4) are sufficiently near g they can be successively joined by
curves which are extremals for the given X. Denote the resulting broken ex¬
tremal by E°. We shall say that (t>) determines the above broken extremal E°.
The value of Jx taken along E° will be denoted by J(v, X).
We shall term J (v, X) an index function belonging to g, to Jx, and to the given
end conditions.
With the aid of the Euler equations one sees that the first partial derivatives
of J(v, X) with respect to the variables vr + i, • • * , v6 are all zero for (v) = (0),
and with the aid of the transversality conditions, that the partial derivatives
with respect to the variables vh • • • , vr, (r > 0) are also zero for (v) = (0).
Thus J(v, X) has a critical point with respect to the variables ( v ) when ( v ) = (0).
We turn now to the terms of the second order of J(v} X). They are obtained
by means of an identity in the variables ( z ) -- (zh • • • , z«), namely
d?
(1.5) Jvavff(0, \)zaZf, = J(ezi, ■ ■ ■ , ez»;X) (a, /3 = 1, • • • , 6; e = 0),
where e is to be set equal to zero after the differentiation.
By the index form associated with g} with the given end conditions and inter¬
mediate manifolds , we mean the form
Q(z, X) = Jvav0( 0, \)zazp (a, 0 = 1, • * • , 5).
The following theorem contains a special application of the theory of the index
form. Its proof is practically identical with the proof of that part of Theorem
8.1 of Ch. II which begins with the paragraph containing (8.2).
Theokem 1.1. In order that an extremal g and set (a) = (0) afford a proper
strong minimum to J relative to neighboring admissible curves satisfying the end
conditions , it is sufficient that g satisfy the transversality conditions , that the Weier-
strass and Legendre S-conditions hold along gy and that the index form Q(z , 0) be
positive definite.
To obtain a representation of the index form in terms of the second variation
we consider the family of broken extremals E° which are £ ‘determined' ' by sets
(v) = (ez 1, • • • , ezi)
in which ( z ) is held fast and e allowed to vary near e = 0. We represent this
family in the form
(1.6) Vi = yi(x, e) (i = 1, • • • , n).
The functions yfx, e) will be of class C 2 for e near 0 and x on the component
extremals of E° between successive points (1.4).
To obtain a more explicit representation of the end points of the extremal
40
THE INDEX FORM
[HI]
segments which make up the extremal E° , it is convenient to use an alternative
notation for the variables (2), namely
(zh ■ • • , Zt) = (uh ■■■ ,ur-,z\, ■■■ ,z\, ■■■ ,z{, ■■■ , zpn).
With this understood the successive end points of the component extremals of
E° are seen to be respectively the points
xl(euu • • ■ , eur), y\(eui9 • • , eur ),
*®(ezi\ • ■ • , cz") Y\{ez ?, • • • , cz«) (? = 1, • • • , p),
x2(euh ■■■ , cur), y*(euu ■ ■ • , cur).
The reader should here recall, in case r = 0, that the symbols .F(a), y*(a) are
used formally for the end points of g.
We turn to our formula for bhk in (3.14), Ch. II. For r > 0 and s not summed,
we set
(1.7) ff9hk = [(/ - Pifp.)x*hk + (fr ~ PifVi)x'hxk +fvlvehy8ik + X*ky*ih) + fPiy*hU],
hy h l,’*',r, i 1,*‘*,ti, & \} &, (°0 (0) i
for (x, ?y, p) on <7 at x — a*. Let
Bik (h, k = 1, • * , n; q = 1, • ■ , p)
denote an expression similar to pflk in (1.7), replacing the derivatives of xs(a)
and y\(a) in (1.7) by the derivatives of Xq(&) and Yq(ff) with respect to the
variables (ft), taking (x, y} p) on g at x = aqy and setting (ff) = (0).
To obtain the second variation of Jx relative to the family (1.6), suppose for
the moment that 6(a) s= 0. Let J\ denote the value of Jx taken along the
gth extremal segment of the broken extremal (1.6) determined by e. For e = 0
we have
<PJ\
de2
d2J\
de 2
: - Plkuhuk 4 / (212 — Xt hrji)dx
B\,z\z *• - Bltiz\z) + £' (2S2 - MiVi)dx,
de 2
^ = PlM - Bv
W + f‘
Ja,
P+1
(2Q — \T}it)i)dx .
(if j \fm*'fU\difk r),
We now restore the function,
0(a) = 9(eui, ■■■ , eur),
DEFINITION OF THE INDEX FORM
41
and combine these results in the formula
(212 — (h, k = 1, • • • , r),
where bhk is given by (3.14) in Ch. II in case r > 0, and is non-existent in case
r = 0.
Here *?*(; r) = y»v(x, 0). Accordingly the functions rji(x) define a broken second¬
ary extremal E with end points and corners respectively at the points
d*Jx
de2
— bhkuhuk +
f.
x = do, • • • y x = ap + j.
This broken secondary extremal is uniquely determined by the set ( z ).
In fact as we have seen in §3, Ch. II (s not summed)
(1.8) i?i(a') = [y\h{ 0) ~ ^(0)y;(a-)]u* + 0 (A = 1, * • • , r)
where the terms in a* are non-existent if r = 0. Similarly for each point x = a,
on f/ we have (7 not summed)
(1-8)' rn-K) = [r?fc(0) - Xtm'Mbl (* - 1, ■ . • , n).
For future reference we write (1.8) and (1.8)' respectively in the forms
Vi ~ <’5^ = 0 (h. = 1 , • • * , r),
(1.8) "
*("*) - = 0 (A- = 1, • • • , n),
where the constants c*h and Cxk are the coefficients of uh and zk in (1.8) and
(1.8)' respectively.
It follows from the non-tangency hypothesis, as we have seen in Ch. II,
that the rank of the matrix || c*ih || is r if r > 0. Similarly it follows from the
fact that the intermediate manifolds Mq are not tangent to g that the rank of the
matrix [ j C9lh 1 1 is //. That is for each intermediate manifold
(1-9) I CU | ^ 0 (iyk = 1, y U).
We draw the following conclusions.
By virtue of the relations (1.8)" an admissible broken secondary extremal (r?)
determines and is determined by a unique set (z).
Theorem 1.2. The index form
Q(Z) X) — J ^aPig(0, \)zazp (a, P = 1, • ,6)
is given by the formula
(1.10) Q(z, X) = bhkuhuk + £ i2uiv , ??') — Xr ur)i]dx {hy k = 1, • • * , r),
42
THE INDEX FORM
[HI]
where
(Zlf f Z&) = ( Ul y y Ur, Z i, , Zn) , 2j, y 2n)
and (rj) is taken along the broken secondary extremal determined by (z) in (1.8)".
We now define what we shall term the special index form.
A particular choice of the “intermediate” manifolds is the set of n-planes
(1.11) x = ax, • • • , x = ap.
A special choice of the parameters (0) on the n-plane x — aq will be the following:
fa = Vi ” (* = h * ‘ * i w).
The relations (1.8)" determining the broken secondary extremal in (1.10) now
consist of the secondary end conditions
(U2) y*i - c*ihuh = 0 (h = 1, • • • , r; i = 1, • ■ • , n)
and the intermediate conditions
(U3) *»(«,) =2? (5 = 1, ,P).
A n index form set up in tins manner will be called a special index form.
Properties of the index form
2. We begin with the following theorem.
Theorem 2.1. The form Q(zy X) is singular if and only if X is a characteristic
root.
The conditions that the form Q(z, X) be singular are that the linear equations
(2.0) G.« = 0 (a = 1, • • • , 5)
have at least one solution ( z ) ^ (0). If such a solution ( z ) is given, we shall
show that the broken secondary extremal E determined by ( z ) affords a char¬
acteristic solution.
Of the conditions (2.0) the first r conditions, taken with the side conditions
(2.1) v’ - c\zh = 0
(h = 1, •
■ ■ , r)>
lead to the conditions
(2.2) Qt/i = 2bhkzk + 2^0,; = 0 .
(h, k = 1, ■
■ ■ , r).
Conditions (2.2) may be written in the form
(2.2)' cLe2, - c\h?\ + bhkzh = 0,
a form identical with the secondary transversality conditions. In satisfying
(2.2)' and (2.1), E satisfies all the boundary conditions that a characteristic
solution must satisfy.
[2]
PROPERTIES OF THE INDEX FORM
43
Let us now examine the geometric meaning of the next n conditions
Qi„ = 0 (a = r + 1, r + n).
These conditions are associated with the corner of E at x — a,. We have found
it convenient to set
(Zr + h ' ) Zr + n) (z i , ' , Zn),
and with this notation in mind we find that
(2.3)
Q'\ = 2 To,. („,„') = 0
L ; dz\ J at
(h 3 = 1, * • * , n)
where (77) is taken on the broken secondary extremal determined by ( z ). From
(1.8)" and (1.9) we see that
g^(Oi)
dz\
rl
* o
From (2.3) we can accordingly conclude that
(2.4) [«,;] = o
a, j = L • • • , n).
(j = L • • • . «).
(Li =!,•••,«),
Conditions (2.4) reduce to the conditions
(2.5) Ja+ = 0
where (x, ?/, p) is taken at x = a\ on g. From these conditions we conclude that
(2.6) [Vi(ai)]ol = 0 (f = 1, • • • , n).
Thus the conditions (2.3) imply that E has no corner at x = a\. Similarly the
remaining conditions imply that E has no corner at the remaining points
X — • • • j X (Ip.
Thus E has no comers at all and satisfies (2.1) and (2.2)'. It is not identical
with the x axis since ( z ) ^ (0). It is accordingly a characteristic solution.
Thus if Q(z, X) is singular, X is a characteristic root.
Conversely let there be given a characteristic solution (tj) satisfying conditions
(4.5) of Ch. II with a constant X, and with r constants (u). Let (z) be the set
which determines this secondary extremal (77), that is, the set (2) which satisfies
(1.8)" with (77). The first r constants in (2) will necessarily be the r constants
(u). Conditions (2.2)', and hence (2.2), are satisfied since (77) is a characteristic
solution. The first r conditions in (2.0) then follow. All conditions such as (2.3)
are satisfied because of the absence of corners on the secondary extremal (77).
Hence all conditions (2.0) are satisfied. Moreover (2) 5^ (0) since (77) ^ (0).
44
THE INDEX FORM
[HI]
Thus Q(z, X) is singular when X is a characteristic root and the proof is complete.
For a given X it is clear that linearly independent secondary extremals that
satisfy the secondary end conditions (2.1) will “determine” and “be determined
Jby” linearly independent sets (2). Since the nullity of the form Q is the number of
linearly independent solutions ( z ) of the equations (2.0), we have the following
theorem.
Theorem 2.2. If X is a characteristic root} the nullity of the form Q(z , X) equals
the index of the root X.
The reader should understand that the number of “intermediate manifolds”
Mg which it is necessary to use to set up the index form Q(z, X) depends upon X.
But a construction valid for a particular X = X0, is also valid for all values of
X < Xo. Whenever we compare index forms for two different values of X we shall
always understand that they are set up with the aid of common intermediate
manifolds M q.
We come to the following lemma.
Lemma 2.1. The form Q(z, X) has the property that
(2.7) Q(z, A') < 0(2, X")
provided ( z ) 9^ (0) and X" < X'.
Let (rj) represent the broken secondary extremal E determined by (2) when
A = A". From (1.10) we have
(2.8) /(. 1, X') - Q(z, X") = (X" - X') j° widx.
From (2.8) we see that
I(r), A') < 0(2, A") (2) * (0).
But from the minimizing properties of the component extremal arcs of broken
secondary extremals,
<2(2, X') ^ /(n, X').
The lemma follows from the last two inequalities.
By the sum of a number of sets (2) will be meant the set (2) obtained by adding
sets (2) as if they were vectors. By a critical set (2) with characteristic root X
will be understood a set (2) ^ (0) at which all the partial derivatives of Q(z, X)
with respect to the variables (2) vanish.
We shall now prove the following lemma.
Lemma 2.2. The index form Q(z, X) is negative if evaluated for a sum (2) 9* (0)
of a finite number of critical sets with distinct characteristic roots each less than X.
Let (2) be the sum. Let X' be the largest of the characteristic roots and (2')
the corresponding critical set. Let (2") be the sum of the remaining critical
sets so that (2) — (2') + (2").
[2]
PROPERTIES OF THE INDEX FORM
45
From the preceding lemma we have
(2.9) Q(z, X) < Q(z, X') (X' < X),
and this inequality proves the lemma if there is but one critical set in the sum,
since the right hand form is then zero.
Now, as a matter of the algebra of quadratic forms,
(2.10) Q(z, X') = Q(z', X') + z'aQJz', X') + Q(z", X') («=!,■■•, 6).
But since (zr) is a critical set for X = X', this equality reduces to
(2-11) Q(z,V) =
We now use mathematical induction, assuming the lemma true for a sum involv¬
ing one less critical set than the original sum. The right hand form is as a con¬
sequence negative. The lemma then follows from (2.9).
Lemma 2.3. The sets (z) in any finite ensemble of critical sets ( z ) with distinct
characteristic roots , arc linearly independent.
Suppose there were such a linear dependence. Let ( z ) be the linear combina¬
tion which is zero. We can regard ( z ) as a sum of critical sets with distinct char¬
acteristic roots. Let ( z' ) and (z") now be defined as in the preceding lemma;
(z') t* (0), and hence (z") ^ (0). Equations (2.10) and (2.11) hold as before.
But the left member of (2.11) is zero since (z) — (0), while the right member is
negative by virtue of the preceding lemma. From this contradiction we infer
the truth of the lemma.
For a fixed number <5 of variables (z) there cannot be more than 5 sets (z)
which are independent. From this fact and the preceding lemma we deduce
the following.
If the index form for X° involves 5 variables (z), there can be at most 6 characteristic
roots less than X°.
From the lemma it also follows that the members of any finite set of char¬
acteristic solutions (77) with distinct roots X are linearly independent.
We now come to a fundamental theorem, Morse [16].
Theorem 2.3. The index of the form. Q(z, X*) equals the number h of characteristic
roots less than X*, counting each root a number of times equal to its index .
To prove the theorem we recall that Q(z, X) will be positive definite for X
sufficiently large and negative. If X now7 be increased, Q will remain non¬
singular except when X passes through a characteristic root Xi. According to
Theorem 2.2 the index qi of such a root equals the nullity of the form Q when
X = Xj. As X increases through \h it follows from the theory of quadratic forms
that the index of Q changes by at most Thus the index of Q(z, X*) is at
most h, the sum of the indices of roots X < X*.
Corresponding to each characteristic root X < X* of index q, there are q linearly
independent critical sets ( z ). According to Lemma 2.2 these sets will make
46
THE INDEX FORM
[HI]
Q(z , A*) negative, as will any linear combination of them not null. But accord¬
ing to Lemma 2.3 the members of any finite ensemble of critical sets with distinct
characteristic roots will be independent. Thus there are h critical sets which
are independent and possess roots A < A*. These h critical sets regarded as
points (z), taken with the point ( z ) — (0), determine an /(-plane in the space ( z ).
On this //-plane Q(z , A*) is negative definite.
It will follow from Lemma 7.1 in §7 that the index of Q(z, X*) is at least h.
But we have seen that it is at most h. Thus the index of Q(z, A*) is exactly h
and the theorem is proved.
We have the following corollary of Theorems 2.2 and 2.3.
Corollary. The index and nullity of the index form are independent of the
number, distribution, and parametric representation of the intei'mediate manifolds
used to define this form, provided the intersections of these intermediate manifolds
vrith g divide g into sufficiently small segments, and provided each intermediate
manifold is regularly represented and not tangent to g.
We shall subsequently make almost exclusive use of the so-called special index
form , defined at the end of §1. The above corollary justifies our use of the
special index form. We have set up the index form in its more general form in
order that we might later establish the geometric invariance of its index.
Conjugate families
3. We shall now obtain certain properties of the differential equations
(3.1) -f 0,' - 0, + X(% = 0 (i = 1, • • • , «).
Corresponding to each solution ij{(x) (class C2), we set
(3.2) U*) = Ml, v’)-
x
From the fact that | fp.p. | ^ 0 along g it follows that each set (x, 77, f) in (3.2)
uniquely determines a set ( x , 77, 77') and conversely. Accordingly two solutions
for which the sets (77, f) are the same at a point x = c are identical. If yji(x) is a
solution of (3.1) and f *(:r) is given by (3.2) it will at times be convenient to speak
of the set rii(x),£i(x) as a solution of (3.1).
If (77, f) and (f 7, f) represent two solutions of (3.1), one has the relation
(3.3) - £i(x)f}i(x) s constant.
In fact the x-derivative of the left member of (3.3) is identically zero, as fol¬
lows with the aid of (3.1). If the constant in (3.3) is zero, then following von
Escherich one terms the two solutions conjugate. See Bolza [1], p. 626.
A first fact to be noted is that if a system S of k independent solutions of (3.1) are
mutually conjugate , then k is at most n.
3]
Suppose k > n. Let
CONJUGATE FAMILIES
47
(3-4)
O' = L • • • 1 n;j = 1, • ■ - , w)
be a matrix of ft solutions of which the jth column gives the ^th solution (77, f )
in the set S. The matrix a is of rank n for every x. In fact, if there were a linear
relation between its columns for x = a, that linear relation would hold identi¬
cally, since the vanishing of a solution (77, f ) at x — a implies the identical vanish¬
ing of the solution (77, f). Now every solution (77, f) of the system S satisfies the
relations
(3.5) - ti}(a)r]i(a) = 0 (i, j = 1, • • - , n)
where a is any particular value of x. We have here n equations in 2 n variables
7}i(a), fi(a). Since the rank of a is n, the variables ^-(a), £\(a) are linearly
dependent on any n independent solutions of (3.5), in particular upon the
columns of a at x = a.
It follows that the solution rjt(x), £\(x) is dependent upon the columns of a,
contrary to the supposition that we had k > n independent solutions. The
proof is now complete.
A system of n linearly independent mutually conjugate solutions will be called
a conjugate base . The set of all solutions linearly dependent on the solutions of a
conjugate base will be called a conjugate family . If the columns of the matrix
(3.4) represent the solutions of a given base, the determinant
D(x) = | ij ij(x) |
will be called the determinant of that base. It is readily seen that the deter¬
minants of two different bases of the same conjugate family are non- zero con¬
stant multiples of one another.
If D(x) vanishes to the rth order at x — a, then x = a will be called a focal
point of the rth order of the given family. It is conceivable that D{x) might
vanish to any order at a point x = a, in fact might vanish identically. The
facts here are given in the following theorem, Morse [13].
Theorem 3.1. If D(x) is the determinant of a conjugate base , the order of its
vanishing atx — a equals the nullity s of the determinant D(a).
We suppose that s > 0 and a — 0.
If 5 is the nullity of D( 0), there exist just s linearly independent solutions of the
conjugate family which vanish at x = 0. Let us take a new conjugate base
1 1 yn{x) 1 1 in which the first 5 columns are these solutions which vanish at x
= 0. If s < n, the last n — s columns of || ya{ 0) || will be of rank n — s, for
otherwise there would exist additional independent solutions of the family
48
THE INDEX FORM
[HI]
vanishing at x = 0, dependent on these last n — $ columns. From each of the
first s columns we can factor out an x and so write
D(x) ~ x*E(x),
where E{x) is continuous in x for x near 0.
(a). I say that E( 0) 3^ 0.
To prove (a) we note that E( 0) is the determinant obtained from | ya(x) | by
differentiating the first 5 columns and then putting x = 0 in all columns. If 6* — n
and E( 0) — 0, one could find a non-trivial linear combination of the columns
of E(0) which would be zero, and which would equal the derivatives ^(0) of
the corresponding linear combination r)i(x) of the columns of |j ya{x) ||. We
would then have
v'dO) = rii{ 0) = 0 (i = 1, • • * , n),
so t hat rji{x) s (), contrary to the fact that the columns of || iji,(x) || are linearly
independent.
Suppose then that s < n and E( 0) — 0. Set.
u%(x) = Cjyri(x) (l = 1, ■ ■ ■ ,n;j= 1, - ■ ■ , s),
Zi(jr) = Chiyth(r) O' = 1, * ■ • , h; h = * + 1, • ■ • , n).
Since E(0) is zero we can determine constants cly ■■ • , cny not all zero, such that
(3.0) u ;•(()) - 0) (i = 1, ••• , n)
as follows from the form of E( 0).
I say that [s(0)] 9* [0]. For otherwise it would follow that x = • • • =
cn = 0, since the rank of the last n — s columns of E( 0) is n — ,s. Hence the
remaining constants ch • • • , r , could not all be zero. Hence (u) ^ (0). But
from the definition of Ui(x) and from (3.6) respectively, we see that if [z(0)]
= [0],
Hi(0) = 0, u • (0) = 0 (i = 1, • * • , n)f
so that uv(x) = 0. From this contradiction we infer that [z(0)] [0].
To return to the proof of (a) we note that Ui(x) and z{(x) are conjugate solu¬
tions. At x = 0 the condition that these solutions be conjugate reduces to
(3.7) fpiPkui(fi)Zi(0) - 0 (t, k = 1, ■ • • , n)
where {x, y, p) is taken at x = 0 on g. With the aid of (3.6), condition (3.7)
becomes
(3.8) == 6*
But ( z ) (0) so that (3.8) contradicts the Legendre ^-condition.
Thus E( 0) 7* 0, and the order of D(x) at x — 0 equals the nullity s of D( 0).
NECESSARY CONDITIONS, ONE END POINT VARIABLE
49
[4J
Corollary. The zeros of the determinant of a conjugate base are isolated and
have at most the order n.
We can now describe the most general conjugate family. According to the
preceding corollary one can always choose a point c at which a determinant of a
base of the family is not zero. One can then always choose a new base such
that
(3.9) vu(c) = 5J (i, j = 1, • • • , ft)
where <5( is the Kronecker delta. Such a base will be called unitary at x — c.
Let
ii f>/(*) ii
be the matrix of the corresponding functions fi(x). In order that the /?th and
kth columns of this new base be conjugate, it is necessary and sufficient that
at x = c
(3.10) Vxktxh — yotik — 0 (i, h, k — 1, • • ■ , n).
Upon making use of (3.9) we find that (3.10) reduces to the conditions
(3.11) = fhk(c).
We have thereby proved that the most general conjugate family F without focal
point at x = c possesses a base satisfying the conditions
(3.12) m,(c) = = f,<(0,
where the values ft ;(c) arc arbitrary except for the condition of symmetry.
We shall term the elements f,,(c) in (3.12) the canonical constants of the family
F at x = a. By virtue of (3.12) these constants uniquely determine the family F.
Necessary conditions, one end point variable
4. We shall here consider the case where the second end point is fixed, while
the first end point rests upon a manifold M, given by the equations
(4.1) x = xl(ah • • * , ar), y* = * • • , ar) (0 ^ r g n)
For r = 0 the set (a) is empty, but as previously we understand symbolically
that for r = 0,
^O) = y\(a) = y,(a').
For r > 0 we suppose that the functions in (4.1) are of class C2 for (a) near (0),
and that for (a) = (0) they give the first end point of g. We suppose 6(a) of
class C2, and define admissible curves and the problem of minimizing J as in
Ch. II.
In case r > 0 a minimizing arc g must satisfy the transversality condition
dd - (/ - PifPi)dxl - fPidy\ = 0
50
THE INDEX FORM
[IN]
regarded as an identity in the differentials dah where (x, y , p) is taken at x = a1
on g. If r = 0 the transversality condition is automatically satisfied. We
suppose that g satisfies the transversality condition.
We now turn to the second variation.
As in §3, Ch. II, we suppose that we have a family of admissible curves
iji = yfc, e)
of the nature of the family (3.2) of Ch. II joining end points determined by
parameters ah — ah(e) (vacuous if r = 0). We suppose that the family gives g
when e = 0. According to Theorem 3.1 of Ch. II the second variation will take
the form
(4.2) J"(0) = bhkuhuk + f“ 2U(V, v')dx = /*(,, u),
where (rj) and the r constants (u) are respectively the variations of e) and
ah(e) (if r > 0) for e = 0, and satisfy the end conditions
(4.3) rt\ - c\huh = 0 (i = 1, • • • , n\ h = 1, • • • , r)
where
(4.4) c\h = y\k(0) - y'i(al)xl( 0).
The constants bhk are given in (3.14), Ch. II.
In addition to the accessory boundary problem previously defined we here
consider a problem to be called the focal boundary problem.
The focal boundary problem shall be defined by the following differential equations
and boundary conditions
(4.5) ' ~ IV, - =0 (* = 1, • ■ • , n),
(4.5) " v\ - c\huh = 0,
(4.5) '" c\h?\ - b^i h = 0 (h, k = 1, • • • , r ).
Let x = a be a point on the x axis distinct from x = a1. Let (tj) be a solution
of the focal boundary problem which vanishes at x — a and which is of class C 2
on the closed interval bounded by x — a1 and x = a. If (77) is not identically
zero between a1 and a neighboring x — a, the point x — a on g will be termed a
focal point of M on g. We extend this definition, including x = a1 as a focal
point of Af on g in case there exists a solution of the focal boundary problem
which vanishes at x = a1, and which is of class C2 but not identically zero neigh¬
boring x = a1.
We shall now derive a necessary condition analogous to the Jacobi condition.
In deriving it we do not assume that the end manifold M is regular, nor do we
need the special assumptions which are made at this stage from the point of
view of the envelope theory.
[5]
FOCAL POINTS
51
Theorem 4.1. If g affords a weak minimum to J in the one-variable end point
problem , there exists no focal point oj the end manifold M at a point x — c on g for
which a1 < c < a2, and at which the Jacobi equations are non-singular .
We have already proved this theorem in case r = 0, that is in case the end
points are fixed. Suppose then that r > 0.
Suppose the theorem is false, and that there exists a set ( ff ) which satisfies the
focal boundary problem with the constants (u) = (u°), and which vanishes at
x — c without being identically zero near x = c. Formula (5.4) of Ch. II with
( v ) = (y) therein and a2 = c, will give us a family of admissible curves e)
joining the points determined by ah = eul on the end manifold, to the point
P on g at which x — c. We extend this family from P to the second end point
of g by following along g.
For the extended family the second variation will take the form (4.2), where
(u) — ( u° ) and (y) defines the curve (X):
(a1 < x ^ c),
(c g x g a2).
Upon integrating the second variation by parts and using the focal boundary
conditions one finds that /*(?7, u°) = 0.
For ( u ) = ( u° ) the curve (X) must afford a minimum to /*(? 7, u°) relative to
neighboring curves of class I) 2 which join the same end points. For in the
contrary case there would exist a curve (77*) of class D2, joining the end points
of (X), and such that
(4.6) I*(y*7 u°) < 0.
One could then use (5.4) in Ch. II to set up a family of admissible curves y^x, e)
of class D 2 for which (77) — (77*) and
(4.7) J"( 0) = /*( 77*, tt°) < 0.
In verifying (4.7) one naturally breaks J up into a sum of integrals between the
corners of the curves yi(x} e). That no contribution to the terms outside the
integral in the second variation is made at the corners is readily seen upon using
(3.11) of Ch. II between corners and summing. But in case g is a minimizing
arc, as we are assuming, (4.7) is impossible.
The curve (X) must then afford a minimum to I*(y, u°) in the fixed end point
problem relative to curves of class D2. Hence (X) must satisfy the Weierstrass-
Erdmann corner condition at x — c in accordance with the remarks following
Lemma 1.2, Ch. I. Exactly as in the proof of the Jacobi necessary condition we
now conclude that fji = 0 near x = c, and from this contradiction we infer the
truth of the theorem.
Focal points
5. Focal points may be regarded as generalizations of centers of principal
normal curvature of a surface. As such they have obvious geometric content.
yt{x) as ffi(x)
Vi{x) = 0
52
THE INDEX FORM
mu
Their theory also serves to unite such diverse elements as conjugate points,
characteristic roots, and the conjugate families of von Escherich. Moreover we
shall see in Oh. IV that the theory of focal points is identical with the theory of
ordinary self-adjoint boundary problems with conditions at one end alone.
We now return to the assumptions of §§1, 2, and 3, namely that g be an ex¬
tremal satisfying the Legendre ^-condition and the transversality conditions.
We also assume that the representation of the end manifold M is regular (r > 0),
and that M is not tangent to g.
From the fact that M is regular and not tangent to g it follows that | c\h | in
(4.5) is of rank r. Hence the parameters (u) can be eliminated from the condi¬
tions (4.5)" and (4.5)'" yielding n linearly independent homogeneous conditions
on the 2 n variables 77 f \ alone. There accordingly exist ?? independent solu¬
tions of the focal boundary problem (4.5) upon which all other solutions are
linearly dependent. Let the columns of a matrix
II »JwOr) II 3 = 1, ,n)
represent n such solutions. WTe continue with a proof of t he following statement.
A ny two solutions of the focal boundary problem are mutually conjugate .
Suppose that (77, f) and (fj, f) represent, two solutions of the focal boundary
problem (4.5), satisfying the boundary conditions of (4.5) with the r constants
( u ) and (u) respectively. If r > 0 we multiply the members of (4.5)'" by uh
and sum. We thereby find that
flifi “ bhkUkUh = 0.
Upon interchanging the roles of the two solutions it appears that
v\l\ ~~ l}hkdkuh ~ fh
Upon recalling that- bhk = bkh we see that
(5.0) fi\t \ - v if! - 0,
so that if r > 0, any t wo solutions of a focal boundary problem are mutually
conjugate. In case r = 0, r)\ = f)\ = 0 and (5.0) is again satisfied.
The statement in italics is thereby proved.
The columns of the matrix || rjij(x) || accordingly form the base of a con¬
jugate family. We shall call 1 7?i/(x) | a focal determinant corresponding to M
and g . The zeros of | r)tj(x) | will be used to define the focal points of M on g .
According to the theory of conjugate families in §3 the zeros of | rjn^x) | are
isolated. As we have seen in §3 a zero x = c of | ruj{x) | possesses an order h
equal to the number y of linearly independent solutions of the focal boundary
problem which vanish at x — c. We term this number y the index of the focal
point x — c. In extending the present theory to the Lagrange problem, the
index of c would be defined as y , not h . The equality y = h does not necessarily
hold in the Lagrange problem.
We shall now give a geometric interpretation of focal points in line with their
15]
FOCAL POINTS
53
classical definition. For the purposes of this interpretation we need to assume
that the functions xl(a), y\{ a) and 0(a) are of class Cz . When this interpreta¬
tion is completed we shall return to the assumption that these functions are of
class C 2.
In case g satisfies the transversality conditions determined by as we are
assuming it does, the manifold M is said to cut the extremal g transver sally.
The following facts flow readily from Theorem 15.1, Oh. V. There exists a
family of extremals which are cut transversally by M at points near gr and which
are representable in the form
(5.1) Z/i = <t> tk, Mi, • • • , M«]
where the functions </>» are of class C 2 in x and (m) for (m) near (0), and give g
for (m) =-' (0). Moreover
yx{al) s 0t-[rr, Mi, * * ' , Mn] (r ~ 0),
(5.2)
?/;(a) = 4>i\x'(ar), au • • • , orr, Mr + o * * * , Mn] (r > 0).
Filially the representat ion is such that t he Jacobian
D(x) =
•• • , 0n)
7>(Ml, * * * , Mn) ’
00 - (0),
has at most an isolated zero at x = a1.
Lemma 5.1. The columns of the determinant J)(x) satisfy the focal boundary
problem.
The lemma is true if r = 0 as we have already seen. We suppose then that
r > 0.
The conditions that the extremals of the family be cut transversally by M
at the point (a) on M may be given the form
(5.3) (/ - PifPi)xlh + fPiy\h - ek = 0 (t - 1, • • • , n; k = 1, • • • , r)
where {x, y, p) is taken at the point (a) on M on any one of the extremals of the
family issuing from that point with (m) near (0).
Let ( u ) be an arbitrary set of n constants and c a parameter neighboring 0
Consider the one-parameter family of extremals
tji = yi{x, e) = <t>{(x, euu ■ ■ ■ , eu„).
{a) = (euh , eur),
x — x'(eui, • • • , eur),
Vi - yi[x'(euu ■ ■ ■ , eur), e],
p. = yi*[xl(euu ■ ■ ■ , eur), e],
(5.4)
If in (5.3) we set
54
THE INDEX FORM
[HI]
then (5.3) reduces to a set of r identities in e. We shall differentiate these
identities with respect to e and set e = 0. In so doing ambiguity will be avoided
if we set
fp.[x, y(x, e), yjx, e)] = F,(x, e).
Differentiating (5.3) with respect to e, we find that for h9 k = 1, • • * , r,
(/ - pJpiWhk^k + teUluk + fv?ly\kUk + j/Pi ^ x[
(5.5) - (; PiFixx\x\uk + p>Fxex\) - j/p. ~~
+ Fxxy\kx\uk + Fity\k + fpiy\hkuk - 9hkuk = 0.
In this result we first cancel the two braces. We then set t?*(x) = yu(x, 0) and
let f i(x) denote the corresponding function QV'. We note that
*
F Jx, 0) =
By virtue of the Euler equations we can also set
(5.6) Fix =
With these simplifications (5.5) takes the form
[(/ - PifpJxL + (/* - PJJxlxl + fv(xjly\k + y\hxi) + fP.y\kk]uk
(5.7)
+ f}(2/a ~ Pixl) ~ OkkUk = 0.
Upon referring to (4.4) and (4.5) we find that (5.7) takes the form
(5.8) fad - bhkuk = 0 {h, k = 1, • • • , r)
where the constants bhk and c\k are those in (4.5).
On the other hand we have the identity
y\(euu ■■■ , cur) = yi[xl(euu • • • , eur), e],
differentiation of which with respect to e leads to the relation
y\hUk = yixxluk -f yie.
Upon putting e = 0 in this relation and recalling the definition in (4.4) of the
constants c\h we see that the variations rji(x) = y^x, 0) satisfy the relations
(5.9) 17 1- - c\huh = 0 (h = 1, • • • , r).
Thus the variations ^(x) and corresponding set ft(x) satisfy (5.8) and (5.9)
combined.
To come to the lemma let (t i)p be the pth column of the determinant D{x)
and let (f)p be the corresponding set (f). We see that the variations (ij)p are
precisely the variations 2/»*(x, 0) of the family (5.4) when the constant up = 1 in
[6]
THE INDEX OF g IN TERMS OF FOCAL POINTS
55
(5.4) and the remaining n — 1 constants (u) in (5.4) are null. With these
constants (u), (r))p and (f)p satisfy (5.8) and (5.9), and the lemma is proved.
We are thus led to the following theorem.
Theorem 5.1. The solutions of the focal boundary problem form a conjugate
family F for which the columns of the jacobian D(x) form a conjugate base. The
jacobian D(x ) is thus a focal determinant belonging to M and g.
The theorem follows at once from the preceding lemma if the columns of
D{x) are independent. But it is known that D(x) has at most an isolated zero at
x = a1, so that its columns must be independent and the theorem is proved.
The index of g in terms of focal points
6. We continue with end conditions of the form
x1 = x‘(ai, • • • , ar), y\ = y\(a,, • • • , a.) (0 S r < n),
(6.0)
x2 = a2, y2i= ijiia2),
where the functions involved are of class C2. In case r > 0 the end manifold
xx(a)y y](a) is to be regularly represented and not tangent to g. We again
consider the functional
j* = e(a) + JxH ^ [/(*> y>y') - \ S (yi ~ yiix))'~\dx-
Corresponding to JF, the end conditions (6.0) and the extremal g, we now set
up the “special index form” Q(z , X) defined at the end of §1. According to
Theorem 1.2,
(6.1) Q(z, X) = bhkuhuk + j * [212 (iy, rj') - \viVi\dx (A, k = 1, • • • , r)
where
(zi, • • • , Zi) = (ui, • • • , ury z\, * • • , zln, • * • , zvly • • * , zvn)
and (?/) in (6.1) lies on the broken secondary extremal whose end points are given
by the secondary end conditions
(6.2)
v\ ~ c)hUh = 0
v • = o,
(i 1> **’ y n} h 1, *** , r),
and whose corners lie at the successive points
(6.3)
X — a\y %it
x — aP} rn(ap>)
We begin with the following theorem.
56
THE INDEX FORM
[III]
Theorem 6.1. The index form Q(z, 0) is singular if and only if the second end
point A2 of g is a focal point of the end manifold M. If Q{z , 0) is singular , its
nullity equals the index of A2 as a focal point of M.
To prove the theorem we note that the addition of the conditions
Vi = o (i = 1, • • • , n)
to our focal boundary problem (4.5) gives a problem By identical for X = 0 with
the accessory boundary problem B\ corresponding to g and to the present end
conditions. Now a necessary and sufficient condition that A 2 be a focal point
of M is that the problem B possess a solution not identically (0). In such a
case the index of A 2 as a focal point of M will equal the index of X — 0 as a char¬
acteristic root of B\, as follows from the definitions of these indices.
The theorem now follows from Theorems 2.1 and 2.2.
Subject to our secondary end conditions at x = a1, namely
(6.4) - c\huh = 0 (h = 1, • • • , r ^ 0),
we can write
bhkUhUt = a% fnWjy — d)i ( ijj = 1, • • * , n),
for suitable choices of t he constant s at7. We then have
(6.5) Q(z, 0) = ai,r,Wi + /* 2fl(n, v')dx
subject to (6.4). We can now prove t he following lemma.
Lemma 6.1. The f unciional
(6.6) airtWj + j ^ 2f2(?7, rj')dx ( h > a1; i, j = 1, • • • , n)
taken over the interval (a1, b) and subject to the conditions
(6.7) Vi(b) = 0 (i = 1, • • • , n)
will be positive definite provided the point x — b is sufficiently near the point x = a1.
We first choose a constant X* so large and negative that the functional
VifflWj + j ^ (2^(17, 1) — Xycoddx 0 iyj = 1, • • • , n)
taken over the interval (a1, a2) is positive definite subject to the conditions
77 i(a2) = 0, provided X ^ X*. For this choice of X the functional
(6.8) dirtWi + J } (2^0b v') - Xyiyddx (i,j = 1, * ■ • , n)
taken over the interval (a1, 5) and subject to the conditions ??*(&) = 0, will be
positive definite for any choice of b such that a1 < b g a2.
[6]
THE INDEX OF g IN TERMS OF FOCAL POINTS
57
We shall now choose the constant b so as to satisfy the lemma.
The problem of minimizing the functional (6.6) subject to the conditions
rn(b) = 0 may be regarded as the problem of minimizing the functional
(6.9)
CiijUiUj +
20(7?, v')dx ,
subject to the end conditions
(6.10)
v\ — y-i, v* = Vi(b) = 0
(* = 1, ' • • > n).
The corresponding accessory boundary problem will then take the form
(6.11) ' 0* - + X* = 0,
(6.11) " = w„ fj = ailui ( t,j =!,•••, n),
(6.11) "' n\ = 0.
All solutions of (6.11)' which satisfy (6.11)* are linearly dependent on the
columns of a matrix
v„(x, A)
fo(*. b)
(h j = 1, • • • . n)
of n solutions of (6.11)' which satisfy the initial conditions
Tli}{n\ X) =
X) — (iij (i, j u).
The determinant D(x, X) = | rj tJ{xy X) ] is continuous in x and X. Moreover
D(a\ X) 1. Hence for a closed interval for X, such as the interval X* g g 0,
there will exist a constant b > a1 differing from n 1 by so little that
D(x, X) 5* 0 (a* ^ x ^ b).
It is now easy to prove that the lemma holds for this choice of b.
In the problem (6.11) there can be no characteristic root less than X*, by virtue
of the choice of X*. Nor can there be any characteristic root X for which
X* g X g 0,
since that would imply thati>(6, X) = 0 contrary to the choice of b. Thus the
problem (6.11) possesses no characteristic roots X S 0. It follows from
Lemma 8.2 of Ch. II that the functional (6,6) is positive definite subject to (6.7),
for the above choice of b.
The lemma is thereby proved.
The following lemma is a first step towards determining the index of the special
form Q(z, 0) in terms of focal points of the end manifold M.
58 THE INDEX FORM [ III J
Lemma 6.2. The index of Q(z , 0) is at most equal to the sum of the indices of the
focal points of M on g between the end points of g.
The set (z) “determines” a broken secondary extremal, the successive ends of
whose segments lie on the n-planes
(6.12) x = a0f • • • , x = ap + i (a0 = a1, ap + i = a2).
For simplicity let us suppose a1 — 0. Let a2 now be decreased to the constant b of
Lemma 6.1, holding a1 — 0, and decreasing the remaining x coordinates x = aq
in the same ratio as a2. For this choice of the n-planes (6.12) we suppose
Q(z , 0) defined and evaluated as before. For this choice of a2 the form Q(z, 0)
will be positive definite.
Now let a 2 increase, the constants ah • • • , ap increasing in the same ratio as
a2, and a 1 remaining null. If a2 thereby coincides with the coordinate x = c
of a focal point of M , the nullity of Q(z, 0) will equal the index k of the focal
point. But as a2 increases the coefficients of the form Q(z , 0) vary continuously.
It follows from the theory of characteristic roots of quadratic forms that the
index of Q(z, 0) will increase by at most k as a2 increases through c. The index
of Q{z , 0) will not otherwise change. Hence as a2 increases from b to its original
value the index of Q(z, 0) will increase by at most the sum of the indices of the
focal points of M on g between the end points of g.
The lemma is thereby proved.
Any curve ru(x) which is of class Dl on (a1, a2) and satisfies the conditions
v\ - c\huh = 0, v] = 0 (h = 1, • • • , r),
with a set of r constants uly • • • , ur will be said to determine a set
(zi, • • • , Z,) = C Ml, ■ ■ ■ , Ur, z\, ■ ■ ■ , z'n, ■ ■ ■ , ZPu • • • , ZPn)
in which the constants z\ are given by (6.3).
We now come to the basic theorem.
Theorem 6.2. The index of the form Q(z, 0) equals the sum of the indices of the
focal points of M on g between the the end points of g .
Suppose the focal points of M on g between the end points of g have x co¬
ordinates
< b2 < • • • <
and that their respective indices are
r i, r2, • • • , ra.
Now the index of the form Q{zy 0) is independent of the number of intermediate
n-planes (6.12) with which one cuts across the x axis provided only that these
n-planes divide (a1, a2) into sufficiently small segments. We can therefore
suppose the n-planes x = aq in (6.12) so placed as to separate the focal
[6]
THE INDEX OF g IN TERMS OF FOCAL POINTS
59
points from one another, and so placed that no n-plane x — aq passes through a
focal point, (q = 1, ... , p).
According to Lemma 6.2 the index v of Q(z, 0) is such that
(6.13) v g ri + r2 + • • • + 7v
We shall prove that (6.13) is an equality.
Corresponding to the focal point at x = bx there are rx linearly independent
secondary extremals
(6.14) h) (j = 1, • * • , r<; i = 1, • • * , a)
which represent solutions of the focal boundary problem and which vanish at
x = bi. From the curves (6.14) for each value of i we now form 7\ new curves
(6.15) g) ( j = 1, * * • , r,; i = 1, • • • , a)
which are identical with the curves (6.14) on the interval (a1, bx ), and are identi¬
cal with the x axis on the interval (6», a2). Let
(6.16) (z)) (j = 1, • * • , rt; i = 1, • • • , or)
be the set (z) “determined” by the curve g). Concerning the sets (6.15) and
(6.16) we shall prove the following:
(a) . The rx + r2 + • * • + ra sets (z) in (6.16) are linearly independent .
(b) . If (v) is taken on any linear combination of the curves (6.15), l(rj, 0) = 0.
(c) . For any linear combination (z) ^ (0) of the sets (6.16), Q(z , 0) < 0.
We shall first prove (a).
Suppose there were a non-trivial linear relation between the sets (z) in (6.16).
Let (77) represent the corresponding linear combination of the curves ^6.15).
We see that (7/) vanishes at each of the points
(6.17) x = a0, • • • , x = ap + 1 (a0 = a1; ap + 1 = a2).
Moreover if x — ak is the last point of the set (6.17) preceding x = ba we see that
on the interval ( ak , b„), (77) represents a secondary extremal (without corner)
vanishing at ak and bff. Hence (77) = (0) on (a*, b0).
Now the only curves of the set (6.15) not identical with the x axis on (a*, bv)
are the curves of the set (6.15) for which i = a , and these curves were chosen
linearly independent. It follows that (77) can involve none of the curves (6.15)
for which i = a. One can now prove in a similar manner that (77) can involve
none of the curves (6.15) for which i = a — 1, and so on down to the curves for
which i = 1. Thus (77) can involve none of the curves (6.15). From this con¬
tradiction we infer the truth of (a).
To prove (b) we represent Z(t;, 0) in the form
I (rtf 0) = bhkUhUh + J ^ 2 {2(77, r\')dx
(6.18)
60
THE INDEX FORM
[III]
as in Ch. II, (4.2). If (77) is any linear combination of the curves (6.15) and
(f) represents the set of corresponding functions f integration by parts in (6.18)
leads to the result
0) “ bhkUhUk + (j = 1, * ‘ * , <r).
i ;
If we make use of the fact that (77) satisfies the secondary end and transversality
conditions, we find that
(6.19) Hv, o) = y, fwj*’.
, '
Now in the neighborhood of x = bj we can write
r)i(x) = Wt(x) + Vi(x),
where Vi(x ) represents a secondary extremal without corner at x — b}} while
W{(x) represents a broken secondary extremal for which
Wi(x) s 0, x ^ bj (i = 1, • • • , n)„
Hence the terat in (6.19) involving 5, reduces to
(6.20) w') J \
We note finally that the secondary extremals ( v ) and (z^), for a: g 6,*, are the
continuations of solutions of the focal boundary problem and hence mutually
conjugate. We see then that the term in (6.20) equals
[wil2,'(t;, t/)J6/ = 0.
Thus 7 (v, 0) = 0 and (b) is proved.
To prove (c) let ( z ) be any linear combination of the sets (6.16), not (0). Let
(77) represent the corresponding linear combination of the curves (6.15), and
(ff) the broken secondary extremal determined by ( z ). According to (b),
7(77, 0) = 0. Now the corners of the curve (77) lie on the n-planes x — bj while
the corners of the curve (fj) lie on the n-planes x — aq. Hence there will be some
extremal segment of (ff) which joins the end points of a portion y of (77), which
portion y is not an extremal segment. Hence
7(U, 0) < 7(77, 0) = 0.
But
Q(z, 0) = 1(f) , 0).
Hence Q(z , 0) < 0 and (c) is proved.
[7]
CERTAIN LEMMAS ON QUADRATIC FORMS
61
To prove the theorem we note that the set of all linear combinations (z) of
the sets (6.16) may be regarded as a set of points on an
(6.21) Tl + r2 + • . . + r,
plane through the origin in the space of the points (z). On this plane Q(z, 0)
is negative definite. According to Lemma 7.1 the index of Q(z, 0) must be at
least the sum (6.21). The theorem now follows from Lemma 6.2.
We have the following remarkable corollary of 'Theorems 2.3 and 6.2. In it
focal points and characteristic roots are counted a number of times equal to
their respective indices.
Corollary 6.1. The number of focal points of the end manifold which lie between
the end points of g equals the number of negative characteristic roots in the correspond¬
ing accessory boundary problem.
We also note the following corollary.
Corollary 6.2. The number of conjugate points of an end point of g between the
end points of g equals the number of negative characteristic roots in the boundary
problem
~ a,; - u,, + - o,
Vv(al) = vi(a2) = 0 (i = 1, • • • , n).
With the aid of this corollary it is easy to prove that there are infinitely many
positive characteristic roots in any accessory boundary problem (Morse [16]).
We shall take this up in Ch. IV in a broader setting.
The two preceding theorems taken with Theorem 1.1 give us the following.
Theorem 6.3. In order that an extremal g afford a minimum to J in our one-
variable end point problem, dt is sufficient that the end manifold cut g transver sally
without being tangent to g,, that the Legendre and Weier strass S-conditions hold
along g , and, that there be no focal points of the end manifold for which a1 < x ^ a2.
Hahn [1] and Rozenberg [1] have made effective use of broken extremals
with one intermediate vertex. They have studied the minimum problem
when the end points of the extremal g are conjugate. They have also deter¬
mined the nullity of the corresponding index form in the case where there
is one vertex and the end points are fixed. The first n conjugate points are
interpreted in terms of classes of broken extremals for which J > J a.
Certain lemmas on quadratic forms
7. The following lemmas on quadratic forms will be extremely useful. The
quadratic forms involved are assumed to be symmetric.
Lemma 7.1. (a) A necessary and sufficient condition that the index of a quadratic
form Q(z) be at least h is that Q(z) be negative definite on some h-plane tt through the
62
THE INDEX FORM
[HI]
origin in the space (z). (b) A necessary and sufficient condition that the index
plus the nullity of Q(z ) he at least k is that Q(z) be negative semi-definite on some
k-plane through the origin.
If the index of Q is p , the form can be carried by a real linear, non-singular
transformation into the form
(7.0) — y\ ----- yl + vl + i + - - - + yh (m ^ p),
where p is the number of variables (z). Suppose that Q is negative definite on
the h- plane x. Now the (p — p)-plane
Vi = * • • = yP = 0
intersects the image of the A -plane x in the space ( y ) in a hyperplane x' of di¬
mensionality at least h — p. If p < h, r' would be more than a point, and it
would follow from (7.0) that Q would not be negative on x'. From this contra¬
diction we infer that the index p is at least h. On the other hand Q is negative
definite on the p-plane
Up + i = ' ’ * = Vp = 0,
where p is the index of Q. Thus (a) is proved .
The proof of (b) is not essentially different and will be omitted.
Our second lemma is the following. Cf. Hilbert and Courant [1]; also Morse
[16], p. 544.
Lemma 7.2. Let Q(z) be a quadratic form in p variables ( z ). Let Qi(v) be the
form obtained by evaluating Q(z) on a (p — p) -plane
(7.1) Z{ = aijVj (i — 1, • , P) j — 1, • f p p).
If the index of Q is k} the index kx of Qi lies between k and k — p inclusive.
If k is the index of Q, there will be a fc-plane x which passes through the origin
in the space (z), on which Q is negative definite. The intersection of x with the
(p — p)-plane (7.1) will be a hyperplane x' of dimensionality at least k — p.
For sets ( v ) (0) corresponding to sets (z) on x', Qi(v) < 0. We must then
have ki ^ k — p.
Since Qi(v ) has the index fci, there exists a fci-plane xi in the space ( v ) on
which Qi(v) is negative definite. When ( v ) is on xi the points (z) given by (7.1)
will lie on a fci-plane xx. On xi, Q(z) will be negative definite. Hence k ^ klm
The lemma is thereby proved.
Lemma 7.3. Let Q'(z ) and Q*(z) be two quadratic forms in p variables (z) such
that
(7.2) Q'{z) = Q"(z) + D{z).
If the indices of Q', Q", D and — D are respectively v'} v" , N and P, then
(7.3) v” - P ^ v9 ^ v" + N.
CERTAIN LEMMAS ON QUADRATIC FORMS
63
The form Q" will be negative definite on a t/'-plane tt passing through the origin
in the space ( z ). There will exist a similar 0 — P)- plane ti on which D ^ 0.
Now tt and ti will intersect in a hyperplane 7r2 of dimensionality at least v" — P.
(We understand that v" — P may be negative, and that 7r2 then reduces to the
O-plane ( z ) = (0).) We see then that Qf(z ) will be negative definite on
.Hence v' ^ v" — P. Upon transposing D(z) to the other side of (7.2), we see
that v" ^ v’ — N.
Relations (7.3) are thereby proved.
Lemma 7.4. Let L(vy w) be a quadratic form in the variables
(»1. ' • • . »r ), (Wh ■ , Wq),
-such that L(vy 0) is non-singular. After a suitable non-singular linear transforma¬
tion from the variables (v, w) to the variables (p, w)y L(vy w) will assume the form
L(v, w) = L(p, 0) + H(w),
■where H{w) can be obtained from L(v, w ) by eliminating the variables (v) by means
of the r equations
(7.4) Lvi(v, w) = 0 (* = 1, • • • , r).
Suppose that
L(vf 0) s aijViVjy an = a,*.
Subject the variables ( v , w) to the non-singular transformation to variables
(p, w) determined by setting
_ L (v, w) = 2(ailp1 + ■■■ +aiTpT) (i = 1, • • • , r),
(7.5)
wk - wk (k = 1, • • • , q).
One can solve equations (7.5) for the variables ( v ) as linear functions of the
variables (p) and (w)f since L(v, 0) is non-singular, and since | a<;- 1 is accordingly
not zero. Under (7.5), L{v} w) will take the form
(7.6) L(v, w) = ctijPiPj + 2 paPiWk + 7 hkWnWk (iyj = 1, * * * , r; h, k == 1, • • • , q)y
where = ajia,ndyhk — y kh-
Suppose now that (p, w) is a second set of variables (p, w) corresponding under
the transformation (7.5) to variables (vyw). If we set up the bilinear form with
the symmetric matrix belonging to L(vy w)y we have
(7.7) ViLv.(vy w) + whLWh{vy w) s 2 anPipj + 2fiikpiWk + 2 pikpiWk + 2y hkWhwk
subject to (7.5). Consistent with (7.5) we now set
Lv.(v} w) = 0, (p) = (0), ( w ) = (0)
if — I j * i
64
THE INDEX FORM
[HI]
keeping (w) and (p) arbitrary. We then see from (7.7) that
2 'fiikptWk = 0
so that fiik ~ 0.
We can thus write (7.6) in the form
(7.8) L(y, w) s L(p, 0) + y hkWhu\
subject to (7.5). If we now reduce L(vt w) in (7.8) to the form H(w) by means
of the conditions (7.4), we must also set (p) — (0) in (7.8) since (7.8) is subject
to (7.5). We thus obtain the identity
(7.9) H(w) = yhkWflwk
from (7.8). Thus
(7.10) L(v, w) = L(p, 0) + //Or)
subject to (7.5), and the lemma is proved.
The preceding lemma will be applied in the following form.
Lemma 7.5. Let L(v, w) be a quadratic form in the variables ( v ) and (w) such
that L(v, 0) is non-singular , and let H(w) be the quadratic form obtained from
L(v, w) upon eliminating the variables ( v ) by means of the r equations
Lrfv, w) = ') (i =■ r).
Then the nullity of the form L(vy w) will equal the nullity of H (w), and the index of
L(v, w) will equal the sum of the indices of the forms L(v, 0) and H(w).
Two end manifolds
8. We have already treated this case under the general theory. We shall
here seek such conditions for a minimum as can be given in terms of the focal
points of the end manifolds together with the usual transversality, Weierstrass,
and Legendre conditions. This problem has been treated by Bliss when n = 1,
See Bolza [1], p. 328. For n > 1 the results now available, as will be seen, are
scarcely predictable from the results for n = 1. The results as here derived
depend upon a use of the index form and a preliminary theory of focal points of
one manifold. Such a theory was given by the author in the Annalen, Morse
[10]. With the aid of these results Dr. A. E. Currier, in a Harvard Thesis,
1930, obtained necessary and sufficient conditions for a minimum. His paper
in the Transactions, Currier [1], modifies his earlier treatment and treats the
parametric case. He restricts himself to the case of n-dimensional end mani¬
folds. In the present section the author treats the case of general end manifolds
in a new manner.
[8]
TWO END MANIFOLDS
65
We suppose that the end manifolds Af1 and M 2 are given respectively in the
forms
(8.1) xl = x\a\, * * • , a1^), y\ = */*(<*}, * ‘ * , al) (0 < n g n),
(8.2) z2 = x\a2ly * • • , a2fj), ?/2 = 2/2(a2, • • • , a2) (0 < r2 g n),
where the functions involved are of class C2 for (a*) near (0), and yield the end
points of g for (a1) — (0) and (a2) = (0) respectively. We suppose that these
end manifolds are regular, and cut g transversally at the respective end points
A1 and A 2 of g, without being tangent to g. We suppose g extended at either
end so as to give an open extremal segment g containing g in its interior. Along
g we suppose that the Legendre ^-condition holds. For simplicity we suppose
that 6 ss 0 in /.
The focal boundary problem (4.5) corresponding to the end manifold M1 will
have boundary conditions of the form
v\ - c\kul = 0,
(8.3)
c\h£i + PLpI = 0 (h, k = 1, ■ , n),
where p\k can be obtained from (1.7) and c\h from (1.8)" upon setting (c*1)
= (a) in the representation of M l. If x — a2 be regarded as the initial end point
of a» extremal segment to the right of x = a2 (that is with x ^ a2), the focal
boundary problem corresponding to Af2 will have boundary conditions of the
form
(8.4)
2
Vi
2 2
CihUh
o,
= o
(//, /r - 1, ■ • ■ , r2),
where filk may be obtained from (1.7) and c\h from (1.8)" upon setting (a2)
= (a) in the representation of M2.
Let Fi and F2 be the conjugate families of secondary extremals satisfying the
conditions (8.3) and (8.4) respectively. Let x = c be any point which is not a
focal point of Ml or M2. Let
(*,i =],■•• > n),
be respectively the two sets of symmetric “canonical constants’’ f\-,(c) of §3
which determine the families F 1 and F2 at x — c.
If g is a minimizing arc in the problem with end conditions (8.1) and (8.2),
no point x = cong between A1 and A2 can be a focal point either of M 1 or of Af2.
In addition to this fact we have the following theorem.
Theorem 8.1. In order that g afford a weak relative minimum to J in the problem
with two end manifolds it is necessary that
(8.5) D(w) = (f i,-(c) - f ^(cflwiWj SO (i, j = 1, • • • , n)
for any set (w) and for any point x = cong between A1 and A2.
m
THE INDEX FORM
[HI]
This theorem will be shown to be a consequence of the fact that the special
index form Q(z, 0) of §1, corresponding to the present problem, cannot be
negative if g is a minimizing arc.
In setting up this index form after the manner of §1 we take the parameters
(a) of §1 as the parameters
(8.6)
(a) = («J, ••• • • - , <).
If g is a minimizing arc there can be no conjugate point of A 1 or A 2 on g between
A1 and A2. Hence in defining Q(z , 0) we need at most one “intermediate”
n-plane, and this we take as the n-plane x = c. The special index form Q(z, 0)
can now be defined as in §1. In it we shall set
(8.7) (z) = ( u\ , - • • , u\t, u\, ■■■ , u\, wh ■■■ , wn)
putting
(8.8) Q(z, 0) = L{u\ u2, w).
We see then from (1.10) that for h} k = 1, • • • , r2 and n, v = 1, • • • , rh
(8.9) L(u\ u2, w) = Plulul - + j 2Sl(v, v')dx.
Here (rj) lies on a broken secondary extremal E with a corner at most at x = c.
The equations by means of which E is determined are given in (1.8)". They
are as follows:
(8.10)
[v\ ~ c\huh
I 2 2 2
l Vi = cikuk
Vi(c) = Wi
(h = 1, ■■■ , r,),
(* = !,•••» rt);
(» = !>•••, ri).
We shall apply Lemma 7.5 to L(ul, u2, w ) to show that the index of L equals
the sum of the indices of the forms
(8.11)
L(u\ u2, 0), H{w),
where H(w ) is the form obtained from L upon eliminating the variables (ul, u2)
by means of the conditions
(8.12)
SL
du\
= 0
(* = !»•••, ri)>
r2).
As a condition precedent to the application of Lemma 7.5 we should know that
the form L(uly u 2, 0) is non-singular. To that end we first note that
(8.13) L(u\ u\ 0) = L(u\ 0, 0) + L(0, u2, 0).
[8]
TWO END MANIFOLDS
67
Now the form L(u\ 0, 0) is the index form associated with the one-variable end
point problem when the end manifold is M1 and the fixed end point is the point
x = c on gy and no intermediate manifolds are employed. Since x = c is not a
focal point of Ml, the form L(ul , 0, 0) is non-singular as affirmed in Theorem
6.1. If we interchange the order of the end points by making a transformation
x = —x, we see in a similar manner that the form L(0, u2y 0) is non-singular.
Thus L(ul , u2y 0) is non-singular and Lemma 7.5 is applicable .
We return to the theorem and note that if g is a minimizing arc the index form
L(u1y u2y w) must have the index zero. Since Lemma 7.5 is applicable we can
infer that it is necessary that H(w) ^ 0.
We shall complete the proof by establishing the identity
(8.14) H(w) m (f‘,(c) - r*,(c))uw/.
Upon using (8.9) and (8.10) we find that the conditions (8.12) can be given
the respective forms
\ + Plkul =0 (h, k = 1, • • • , r,),
(8.15)
c*hti + Plkul = 0 (h, k = 1, , r2).
Since L(ul, u2f 0) is non-singular the conditions (8.12) determine the variables
u\, u\ as linear functions ul(w), u\(w) of the variables (w). For these variables
ul(w)y u\(w) the variables rj * can be taken so as to satisfy (8.10). Thus for a
given set (w)y the conditions (8.10) and (8.12) can be satisfied simultaneously.
Since (8,15) then holds on the corresponding broken extremal Ewy we see that the
two segments of Ew belong respectively to the two families F i and F2.
Now a member ( rj ) of the family Fx which satisfies the conditions
(8.16) ??i(c) = Wi {i = 1, • * • , n)
determines a set f»(x) for which
(8.17) ^(c) = (v) in F}.
A member of the family F2 which satisfies (8.16) determines a set f<(: r) for which
(8.18) f.(c) = f 2ij(c)wj, (rj) in F2.
We shall represent H(w) by means of the right member of (8.9) noting that
(ri) therein satisfies (8.10) and (8.15). If we then integrate by parts over the
intervals (a1, c) and (c, a2) respectively, we find that for the set rn(x) in (8.9)
and the corresponding set f*(z)
h(w) =
But we are concerned with a broken secondary extremal Ew whose two compo¬
nent extremals satisfy (8.17) and (8.18) respectively, as well as (8.16). Hence
H{w) = [f \3(c) - f^-(c)K^y,
as was to be proved.
68 THE INDEX FORM [ III ]
We have previously noted that it is necessary that II (w) ^ 0 if g is a minimiz¬
ing arc. The theorem follows directly.
We can now prove the complementary theorem.
Theorem 8.2. In order that g afford a proper , strong , relative minimum to J
it is sufficient thai the end manifolds M1 and M 2 cut g ti ansversally without being
tangent to g} that the Weierstrass and Legendre S-conditions hold along g, that there
be no focal points of Ml or M2 on g between M 1 and M 2, and that the form D(w) of
Theorem 8.1 be positive definite .
We first note that there can be no conjugate point of A1 or A2 on g at a point
x — c between A1 and A 2. For under the conditions of the theorem a segment of
g between x = a1 and x = x0, with c < x0 < a2, will afford a minimum to J in
the problem with one end point variable on M 1 and the other fixed at x — x0 on g ,
as we have seen in Theorem 6.3. By the Jacobi necessary condition in the
fixed end point problem no such conjugate point as x — c can then exist.
We can nowT set up t he index form L(ul, u 2, w) as in the preceding proof with an
intermediate n-plane x — c. We note that the forms L(ul, 0, 0) and L( 0, u2, 0),
interpreted as index forms as in the preceding proof, must be positive definite,
since there are no focal points of M1 for which a1 < x S c, or of M2 for which
c £ x < a2. Moreover we have seen in (8.14) that II (w) =* D{w)y so that
II (w) is positive definite, as well as L(ul, u 2, 0).
It follows from Lemma 7.5 that L(ul , a2, w) is positive definite. The theorem
follows from Theorem 1.1.
From this point on we shall assume that there are no focal points of M 1 or M2
on g between A1 and A 2. In counting focal points we adhere to the convention
that a focal point is to be counted a number of times equal to its index. More¬
over we shall say that a point at which x > x0 lies to the right of a point at which
x = Xo.
We then come to the following lemma.
Lemma 8.1. If the focal points of Ml and M2 for which x ^ a2 are respectively
counted in the order of increasing xy then a necessary condition that
(8.19) (f <,(c) - Ci,{c))wiWj ^0 (a1 < c < a2),
is that the kth focal point of Ml on g lie to the right ofy or coincide with , the kth focal
point of M2,
Let x = c be a point on g for which c > a2. Consider the functional
Jl = — + J' Q(y, y')dx (a1 < c < a2 < c),
taken along curves of class D1 wrhose first end point lies on the n-plane x = c at
the point
(8.20)
x — c,
yi = on
(i = 1, • • • , n)
[8]
TWO END MANIFOLDS
69
and the other end point is found at the point
(8.21) x - c, j/i = 0.
Consider also a second functional
j, = r-,(c)«,a, + j' n(y> yl)dx
subject to the same end conditions. The x axis between x = c and x = 6
inclusive, will be an extremal g° relative to both functionals and will be cut
transversally by the n-plane x = c. The focal boundary problem (4.5) corre¬
sponding to these two functionals, to the extremal g°, and to the end manifold
(8.20), will possess boundary conditions of the respective forms
(8.22) v <(c) = Ui, fi(c) = ri,(c)n,- (i = 1, • • • , n),
(8.23) v i(c) = Ui, f,(c) =• rt,(c)u,.
But these two focal boundary problems are seen to define precisely the con¬
jugate families F1 and F 2. Thus the focal points on the x axis, of the n-plane
x = c, relative to the functionals J 1 and J 2 have the same x coordinates and in¬
dices as the focal points of M1 and M2 relative to J on the extremal g.
Relative to J 1 and J2, the extremal g°, and t he end conditions (8.20) and (8.21),
we now set up special index forms Ql(z, 0) and Q2(z, 0) respectively, using the
same intermediate n-planes in the two cases. We then have two formulas
(8.24) Q’(z, 0) = + J' 2tt(Vt v')dx ( i , j = ;1, • ■ • , »; < = 1, 2),
where we set the first n variables in the set ( z ) equal to n variables (u), and where
(77) is taken on the broken secondary extremal determined by (z). From (8.24)
we see that
(8.25) Ql(z, 0) - Q2(z, 0) = - ft, -(c) ]uiUj.
From (8.19) we see then that
Q\z, 0) ^ 0).
The index of Q2(z, 0) must then be at least as great as that of Qx(zf 0). This
means that the conjugate family F2 must have at least as many focal points
between x — c and x = c as does jFV
The lemma follows directly.
From this lemma and from Theorem 8.1 we infer the following.
Theorem 8.3. In order that g afford a weak relative minimum to J it is necessary
that there be no focal points of Ml or of M 2 on g between Ml and M2, and that the
kth focal point of Ml on g to the right of A1 lie to the right of or coincide with the kth
focal point of M 2 to the right of A2.
70
THE INDEX FORM
[IH]
We shall now prove the following lemma.
Lemma 8.2. In order that the difference form
D(w) - [fi,.(c) - (a1 < c < a2)
of Theorem 8.1 &£ positive definite it is sufficient that theie he no focal points of Ml
or of M2 on g between Ml and M2, and that on some segment of g for which c < x
< c there be n more focal points of M2 than of M1,
We consider again the forms Ql(z, 0) and Q2(z , 0) set up in the proof of Lemma
8.1. We write the relation (8.25) in the form
QK*, 0) = Q2(z, 0) + D(w),
where ( w ) gives the first n of the variables ( z ). Let the indices of —I), Q1, and
Q2 be respectively P , v', and v". According to Lemma 7.3,
(8.26) P ^ v" - v'.
Now if there are n more focal points of M 2 than of M 1 for which c < x < c, we
must have v" — v' — n. We then see from (8.26) that
P — n.
Hence D(w) is positive definite, and the lemma is proved.
We see incidentally from (8.26) that v" — v' can be at most n, that is there
can be at most n more focal points of M 2 than of M1 for which a1 < c < x < c.
From this lemma and from Theorem 8.2 we obtain the final result.
Theorem 8.4. In order that g afford a proper , strong , relative minimum to J it is
sufficient that the end manifolds Ml and M2 cut g transver sally without being tangent
to g , that the Legendre and Weierstrass S-conditions hold along g, that there be no
focal points of Ml or M2 on g between M1 and M2, and that on some closed extremal
extension of g on which the Legendre S-condition holds there exist a segment a2 g
x < con which there are n more focal points of M 2 than of MK
The r61es of M 1 and M2 can be interchanged in an obvious manner.
Periodic extremals, a necessary condition
9. In the following three sections we shall suppose that the integrand /(x, yf y ')
as well as the functions y%(x) and y\(x) have a period a> in x. For simplicity we
set 6(a) = 0 and take a1 as 0 and a2 as o>.
Our end conditions here have the form
<9.») X‘ " 0> V‘‘ ‘ “*
X* = CO, y\ = CLi (i = 1, • • • , n).
The corresponding secondary end conditions become
x1 = 0, v) = Ui,
= u, ij * = ut
(9.1)
(t = 1, • • • , n),
71
[ 10 ] THE ORDER OF CONCAVITY
while the secondary transversality conditions reduce to the conditions
rl - r • = o.
The second variation I(t), X) of Jx takes the form
(9.2) I(n, X) = [2Q(tj, t}') — \tuTn]dx.
Two points in the (x, y) space whose x coordinates differ by w and whose co¬
ordinates ( y ) are the same will be called congruent points. We make a similar
convention for the space ( x , 77).
If g affords a minimum to J relative to neighboring curves of class D1 that
join congruent end points, it is necessary that there be no periodic solutions of
the accessory differential equations for which X < 0 and (77) ^ (0), as we have
already seen. Moreover if x0 is any value of x, it is also necessary that there be
no conjugate point of x between x0 and x0 + The following theorem contains
still another necessary condition.
Theorem 9.1. If g affords a weak minimum to J relative to neighboring admis¬
sible curves joining congruent end points , it is necessary that
(9.3) rn(x0 + w)f*(x0 + w) — 77l-(xo)f»(x0) ^ 0
for every solution of the Jacobi equations which is of class CJ and joins congruent
points on the n-planes x = 2*0 and x = x0 + w respectively .
Suppose that 77 t-(x) is a solution of the Jacobi equations of the nature described
in the lemma. Regard this solution 77 »(x) as defined merely on the interval
(x0, x0 + oj). Let the functions yfx) now be defined at all remaining points
x by the condition that rjt(x) have the period w. The curves
yfx, e) = yfx) + ^-(x) (0 S x g w)
will then form a family of admissible curves joining congruent points in the
(x, y) space. For this family the second variation of J integrated by parts in
the usual way will reduce to the left member of (9.3), at least if x0 = 0. If
x0 0 an obvious use of the periodicity of yjx) leads to the same result.
But for a minimizing arc g , /"(0) cannot be negative. Hence (9.3) must hold
and the theorem is proved.
The order of concavity
10. We continue with the periodic extremal g of the preceding section. Along
g we now assume that the Legendre ^-condition holds.
We have already determined the index of a periodic extremal in terms of the
characteristic roots of the accessory boundary problem. In the next section we
shall give another mode of evaluation of this index in terms of conjugate points
and a new numerical invariant. This new invariant will now be defined.
72
THE INDEX FORM
[III ]
For each value of X near 0, let
(10.1) || pai: x, X) 1|, || qn(x, X) || (i, j = 1, ■ , n)
be respectively n-square matrices whose columns are solutions of the accessory
differential equations set up for (9.2). Let
IlftOr, x)||, II fly Or, X) II
be respectively the matrices of the corresponding sets f We now suppose that
these solutions satisfy the initial conditions
8\ 0
| 0 5J
Now for a given X any secondary extremal can be given the form
(10.3) vi(x) = bjpa(x, X) + X)
where the 6/s and c/s are constants. One sees that a necessary and sufficient
condition that some solution be periodic and not identically zero is that
Pij( a>, X) - $5, Qi/u, X)
(10.4) - 0.
X )y f»,(^> X) ~
If the condition (10.4) holds for X = 0, we term g degenerate. We shall assume
throughout this section that g is non-degenerate.
Let F\ be the family of those secondary extremals which join congruent points
on the n-planes x — 0 and x — u for the given X. We seek a base for the family
Fx. We shall restrict ourselves to values of X near 0. The conditions that (*?)
in (10.3) define an extremal of F\ are that
(10.5) l>jpa( co, X) + crfi/w, X) - bi = 0.
The matrix of the coefficients of the constants ( b ) and (c) is
(10.6) 1 1 X) - <>', <?i;(o>, X) || .
By virtue of our assumption that g is non-degenerate this matrix will be of rank
n for X = 0, and hence of rank n for X sufficiently near 0.
For X sufficiently near 0 all solutions ( b , c) of (10.5) will be linearly dependent
on n particular independent solutions of (10.5), and these solutions can be so
chosen as to vary continuously with X. By virtue of (10.3) these n particular
solutions of (10.5) will define n particular independent solutions of the Jacobi
equations upon which all solutions in the family will be dependent. We
represent these solutions by the columns of the matrix
(10.2)
Vij( 0, X),
<7 >;(0) X)
r?,-(o, x),
n,(o, x)
(10.7)
1 1 x) 1 1
THE ORDER OF CONCAVITY
73
[10]
and let
(10.7) ' iu;,(*,x)ii
be the matrix of the corresponding sets
Members of the family Fx can be represented in the form
(10.8) Vi = Zik(x, \)wk, ?{ = f *ik(x, \)u\ ( i , h, k = 1, • • • , n),
where the wf s are constants. If the second variation I(rj , X) be taken along
the curve (10.8), we find that
(10.9) i(v, x) = o-
If one uses (10.8), the right member of (10.9) reduces to a quadratic form
ahk(\)whwk = D(w , X)
in which
(10.10) ahk(\) = zik(0, \)[>'A(:r, X)]“ n).
We shall term J)(w, X) the general difference form corresponding to the segment
(0, o>) of the x axis. It is defined only for X near 0. WTe shall establish three
properties of this form.
I. The form D(w , X) is symmetric.
To see this recall that the hth and &th columns of the base (10.7) satisfy the
conditions
~ ziktih s constant
identically in x for each X. Upon successively substituting x = 0 and x - co
in this identity one finds that ahk = akh as required.
For a particular value of X, say X°, it may be possible to set up a special base
(10.11) 1 1 *!,(*) 1 1, II r!, toll.
for the family of secondary extremals Fx 0 using some special definition not
applicable for all values of X near 0. For such a special base the form
J)°(w) = a°„ w»n\ (m, v = 1, • • • , n),
in which
al = ^„(0)[r^to];,
will be called the corresponding special difference form.
We shall now prove the following.
II. For X = X° the index of the general difference form D(w , X°) equals the index
of any special difference form D°(w) set up for X = X° corresponding to a special
choice of base for the family Fxo.
74
THE INDEX FORM
[HI]
Between our bases we necessarily have a relation for X = X° of the form,
z* = (*, m, A = 1, • • • , n),
t <* = f (", A = 1, • • • , n),
where 1 1 1 1 is a non-singular n-square matrix of constants. If we make use
of the definitions of akk and we find that
a**(X°) = crkal,c„k.
According to the theory of quadratic forms the indices of the forms D(w> X°)
and D°(w) must then have the same values, and II is proved.
III. The nullity of the general difference form D(w, 0), evaluated for X = 0r
equals the index of x — w as a conjugate point of x = 0.
To establish this fact we first note that
i rih( co, o) - r :*(o, o) 1 5* o.
Otherwise one could readily obtain a periodic solution of the Jacobi equations
not (tj) s= (0). If we now turn to the definition of a^( 0) in (10.10), we see that
the nullity of | ahk( 0) | equals the nullity of j zik( 0, 0) |. The latter nullity is
seen to be equal to the number of independent solutions of the Jacobi equations
which vanish at x — 0 and x = a>, that is, the index of x = w as a conjugate
point of x = 0.
Statement III is thereby established.
We shall term the index of D(w, 0) the order of concavity of the segment (0, co)
of the x axis.
The justification of this definition will appear later. In it we have associated
the form D(w , 0) with a particular segment (0, a?) of the x axis. This is neces¬
sary. In fact if one should change the origin to some other point x = x0, the
index of the new form D(w, 0) would not necessarily be the same as that of the
old, as simple examples would show.
If x — 0 and x = co are not conjugate for X = X° a special difference form D°(w)
can be set up as follows. As the base (10.11) we can take a set of solutions of
the Jacobi equations such that
(10.12) *U«) = z°y( o) = (i, j = 1, • • • , n).
The corresponding ' ‘special difference form’7 D°(w) then reduces to the form
(10.13) D°(w) = w,w,[fi,Cr)]“.
We shall use this form in the next section.
The index of a periodic extremal
11. We continue with our study of a non-degenerate periodic extremal along
which the Legendre S-condition holds. We set up the special index form Q{zr X)
of §1 corresponding to end conditions of the form (9.0).
[11]
THE INDEX OF A PERIODIC EXTREMAL
75
We find that
(11.1) Q(z, X) = J [2 7)') — \rjiTji]dx
where (tj) is the broken secondary extremal “determined” by ( z ).
We shall now prove the following theorem.
Theorem 11.1. If g is non-degenerate, its index will equal the number of con¬
jugate points of x — 0 on the interval 0 < x ^ a>, plus the order of concavity of the
segment (0, o>) of the x axis. (Morse [7, 17].)
We distinguish bet, ween two cases.
Case I. The points x — 0 and x — co are not conjugate. In this case the
special difference form D°(w) can be set up for X = 0 as at the end of §10. By
virtue of §10, II, the order of concavity of (0, a>) will then equal the index of
D°(w).
We shall base our proof of the theorem under Case I upon Lemma 7.5. To
apply this lemma it will be convenient to denote the first n of the variables ( z )
in Q(z , X) by (wu • • * , wn) and the remaining 8 — n variables ( z ) by
(in, • • • , vs _ n). We then write Q(z , X) as a form
Q(z, X) = IS(w, v).
We note that the form />°(0, v) is non-singular since L°(0, v) is the special index
form associated with the fixed end point problem, and since the end points of g
are not conjugate. According to Lemma 7.5 the index of Q(z , 0) will equal the
index of L°(0, v) plus the index of a form II (w) obtained from L°(w, v) by
eliminating the variables (y) by means of the conditions
(11.2)
dL°(w , v)
dVj
(j = 1, * * * , & ~ n).
To interpret the conditions (11.2) we turn to (11.1). In (11.1), (77) and (f)
must be taken on the broken secondary extremal A7 determined by (; z ) — (w, v).
With the aid of (11.1) one sees that the conditions (11.2) reduce to a set of n
conditions of the form
(11.3)
(*=!,••■, n)>
one set at each corner x — aq of E. The conditions (11.3) and hence (11.2)
imply the absence of corners on E .
We wish to determine the index of II {w). Now subject to (11.2), // (w) =
Q(z, 0) by definition of H(w). From (11.3) we see in addition that subject to
(11.2)
(11.4)
II(w) = Q(z, 0) =
70
THE INDEX FORM
[HI]
Here (j?) and (f) are on the secondary extremal determined by (z). But by virtue
of (11.2), (z) is determined by ( w ), so that (y) and (f) in (11.4) must be on the
secondary extremal Ew which joins the points
x = 0, yi = Wi,
X = (x), rji = Wi .
But the functions (77), (f) on Ew can be represented in terms of the special base
defined at the end of §10 as follows:
Vi = Ak(x)wk,
i =
(i, h, k = 1, • • • , n).
We see then that H(w) in (11.4) reduces to the difference form
H(w) - M>Aw*[fXt(a:)]" = D°(w)
of (10.13). Thus the index of H(w) equals that of D°(w).
According to Lemma 7.5 the index of Q(z, 0) equals the index of L°(0, v) plus
the index of H(w). But the index of L°(0, v) is the number of conjugate points
of x = 0 on the interval 0 < x S o>} and the index of H(w) is the index of D°(w)f
that is, the order of concavity of the segment (0, a>). The theorem is accordingly
proved in Case I.
Case II. The point x = a> is a conjugate point of x = 0 of index p (X = 0).
This case can be treated as a limiting case of the preceding.
For X ^ 0 but sufficiently near 0, x = co will not be conjugate to x = 0, for
otherwise the characteristic roots in the fixed end point problem would not be
isolated. Let a be the number of conjugate points of x = 0 preceding x = o>
for X = 0. For X < 0, but sufficiently near 0, there will be a conjugate points of
x = 0 preceding x — w, while for X > 0 there will be a + p such conjugate points.
This follows from the fact that the number of conjugate points of x = 0 on the
interval 0 < x < co for a given X equals the number of characteristic roots less
than X in the fixed end point problem.
Let u~, u°, and u+ be respectively the indices of the general difference form
j D(w, X) of §10 for X < 0, X = 0, and X > 0, with X near 0. For X > 0 but
sufficiently near zero, g regarded as a periodic extremal of Jx comes under Case I.
For Case I the theorem is already established. Hence if r is the index of Q(z, X)
we have
(11.6) t = a + p +
Similarly for X < 0 (r is unchanged)
r == cr + U~.
Hence
(11.7)
p — u~ — tC.
[11]
THE INDEX OF A PERIODIC EXTREMAL
77
But according to III in §10, the nullity of D(w , 0) is p, while (11.7) tells us
that the index of D[w, X) decreases also by p as X increases through 0. It follows
from the theory of characteristic roots of quadratic forms that
u° = iC.
Hence (11.6) gives the result
r = a + p + u°.
The theorem is thereby proved in Case II.
We shall make use of Theorem 11.1 to obtain sufficient conditions for a
minimum. It will be illuminating however first to note the following special
necessary conditions. If g affords a minimum to J, it is necessary that the form
Q(z, 0) have the index zero. If then g is non-degenerate and affords a minimum
to J, it follows from Theorem 11.1 that x = 0 cannot be conjugate to x = w and
that the order of concavity of (0, a>) must be zero.
The way is thus prepared for the following corollary of Theorem 11.1.
Corollary. In order that a non-degenerate periodic extremal g afford a proper ,
strong, relative minimum to J , it is sufficient that the Legendre and Weierstrass
S-c,onditions hold along g, that there be no conjugate point of x = 0 on the interval
0 < x ^ w, and that the order of concavity of the interval (0, o>) be zero.
The case n = 1. In this case we can obtain a very explicit determination of
the order of concavity of the segment (0, u). For n — 1 there is but one variable
yi or pi so that subscripts can be dropped.
There are two cases according as x = 0 is or is not conjugate to x = w.
Case I. Suppose first that x = 0 is not conjugate to x = co. The special base
II z°i,(x) | [defined at the end of §10 reduces here to a single solution rj(x) such
that
r/(w) = 1,(0) = !•
If f(x) is the corresponding function the special difference form (10.13)
reduces to the form
(?(») ~ f(0))w?.
But
f(«) - f(0) = fM( ») - n'(0))
where /pp is evaluated at x = 0 on g. The order of concavity of (0, ») is thus
1 or 0 according as
??'(a>) < n , (0)
or
Tj'(w) > *?'(0).
78 THE INDEX FORM [ III ]
In the first case we say that the segment (0, o>) is relatively concave , in the second
relatively convex (Morse [3] p. 239).
Case II. There remains the special case in which x — 0 is conjugate to x = w.
According to III, §10, the nullity of D(w, 0) will then be 1. Hence D(w , 0) s= 0
and the order of concavity is zero.
The index of a non-degenerate periodic extremal g in the plane can accordingly
be evaluated as follows: n = 1.
(A) . Let m be the number of conjugate 'points of x = 0 on the interval 0 < x < cj.
If x — 0 is not conjugate to x — w, the index of g is rn or rn + 1 according as (0, co)
is relatively convex or concave . If x — 0 is conjugate to x = co, the index of g is
m "T 1.
We shall use the preceding to establish a result of importance in Ch. IX.
(B) . Let g be a non-degenerate periodic extremal on which a point x is never con¬
jugate to the point x + co, and on which there are rn conjugate points of the point
x = 0 on the interval 0 < x < co with rn > 0. The index of g will then be m or
rn + 1 according as rn is odd or even .
Since the point x = 0 is not conjugate to the point x = to there exists an
secondary extremal E on which
*?(0) = ??0) = T
Let E' be a secondary extremal obtained from E by replacing each point (r, 77)
on E by the point (x + co, 77). We see that E and Ef intersect at (co, 1). More¬
over E and E' will not be tangent at (co, 1), since E would then represent a
periodic extremal contrary to the hypothesis that g is non-degenerate. As x
increases through w, E thus crosses E' at (to, 1). We see that the segment (0, co)
is relatively concave, or relatively convex, according as E enters or does not enter
the region between E' and the x axis when E crosses E' at the point (a?, 1), with
increasing x.
To prove the theorem we have merely to show that the segment (0, co) is
relatively convex if m is odd and relatively concave if m is even.
We consider the case where rn is odd, say m = 2r — 1 with r > 0. In this
case we shall prove that as x increases through co, the extremal E cannot enter the
region between Ef and the x axis at the point (co, 1).
If we use the Sturm Separation Theorem, and compare E with a secondary
extremal which vanishes at x = 0, but on which 77 is not identically null, we see
that on E} 77 must vanish 2 r times on the interval 0 < x < o>.
Let x — a be the first zero of 77 on E following x = 0. The first conjugate
point of x — 0, following x = 0, must follow x = a, as the Sturm Separation
Theorem shows. Hence the first conjugate point of x = following x =
must follow x — a + a>. By virtue of the Sturm Separation Theorem, E cannot
then intersect E ' on the interval
(11.8)
w < x ^ a + w.
[11]
THE INDEX OF A PERIODIC EXTREMAL
79
Moreover E cannot intersect the x axis on the interval (11.8). For that would
mean that the point x = a had 2 r conjugate points on the interval
(11.9) a < x g a + «.
But the point x = 0 has 2 r — 1 conjugate points on the interval 0 < x < and
upon continuously varying a from 0 to its given value, the number of conjugate
points on the interval (11.9) would remain 2r — 1 since no point x is conjugate
to the corresponding point x + co.
Thus E can intersect neither E' nor the x axis on the interval (11.8). It
follows that E cannot enter the region bet ween E' and the x axis at the point
(co, 1). Hence if m is odd the segment (0, co) is convex and the index is m.
The proof of the lemma in the case m is even is similar.
For the general theory of the minimizing periodic extremal prior to the work
of the author, the reader is referred first to Hadamard [1], p. 432. The Poincare
necessary condition that there be no pair of conjugate points on a minimizing
periodic extremal is here derived together with other conditions bearing on a
minimum in the plane. Carathfodory [2] has considered periodic extremals in
w-space. Among other results he has shown that the Poincar6 necessary con¬
dition does not hold. Hedlund [1] has shown that the Poincar<$ condition does
not hold even for surfaces in the non-orient able case. Further references to
papers on periodic extremals will be given in Ch. IX in connection with the
theory in the large.
CHAPTER IV
SELF-ADJOINT SYSTEMSf
That the calculus of variations had much to do with the theory of separation,
comparison, and oscillation theorems was evident even in the papers of Sturm.
Certain aspects of this fact have been strikingly brought out by Hilbert and
Courant [1], But the nature of the results so far obtained calls for the setting
up of a general framework and theory for such problems. The present chapter
aims at such a theory. Although the results are confined to the case of a system
of second-order self-adjoint differential equations with self-adjoint boundary
conditions, yet they are capable of a much broader development. In particular
one could consider such systems of second-order and linear differential equations
as appear in the accessory differential equations of a Lagrange problem (Morse
[16]). In particular by a reduction to a Lagrange problem the baffling case of
the general even-order, self-adjoint, ordinary differential equation can be
successfully treated. (Results not yet published.)
Starting with a new parametric representation of self-adjoint boundary con¬
ditions, comparison theorems are classified in a general way and new numerical
invariants are introduced. A mode of proof of the existence of characteristic
roots is developed which for the case at hand is more powerful than any hitherto
developed. In particular one may recall that the methods of integral equations
depend in general upon the fact that the parameter enters linearly and analyti¬
cally. Such restrictions are unnecessary here. Missing oscillation theorems
for general boundary conditions are here obtained. Cf. Hickson [1]. Finally
the theory of boundary problems, self-adjoint at one end point, is shown to be
identical with the theory of focal points, thus giving this class of problems a
geometric setting.
Since these Lectures were given, Dr. Kuen-Sen Hu [1] has generalized the
results previously published by the author, Morse [10, 16], to a form of the Bolza
problem with somewhat less restrictive hypotheses. In the present chapter we
make use of a classification of separation, comparison, and oscillation theorems
which enables us to go deeper into the questions involved. The generalization
of our theorems to the Lagrange problem under suitable normalcy conditions is
obvious.
Among the earlier papers one may refer to Bliss [9], Plancherel [1], Richardson
w.
Self-adjoint differential equations
1. Consider a system of n differential equations of the form
(1.0) Li(rj) = Atp 7, + H i jijj + Ciflj = 0 (i, j = 1, • • • , n)
fThis chapter can be omitted by the reader interested chiefly in the theory in the large.
80
[1]
SELF-ADJOINT DIFFERENTIAL EQUATIONS
81
where x is the independent variable and A a, jB<, and C<; are continuous functions
of x on the interval a1 ^ x ^ a2. If A »; is of class C1, these differential equations
can be written in infinitely many ways in the form
(1.1)
j “b bijvi) (.CijVj ~b diflj) — 0
where a,y and are of class C 1 and ca and di9 of class C° in x. The system Lfirj)
will be unchanged as differential conditions if we replace c*,, and 6tJ respec¬
tively by
difix) + g'ijix),
(1.2) ctfix) + ga(x)9
bifix) + gij(x),
where gifix) is an arbitrary function of x of class C1. We term such changes
admissible modifications of (1.1). In particular we can use this arbitrariness of
the coefficients to make di fix) an arbitrary set of symmetric elements
(1.3) difix) = djfix)
of class C1. We can then still add an arbitrary constant to c,*, and
We shall now use a definition of self-adjointness which will not require the
assumption of further differentiability of the coefficients in (1.1). To that end
let
(1.4) M(U, V , U v') = aijUiVj + PijUiV) - 7 ijUiVj ( hi = 1> * * * > **)
be a bilinear form in which the coefficients a»y, Pa, and ya are continuously
differentiable in x. We shall say that the system (1.1) is self-adjoint if there
exists a bilinear form M such that the condition (Davis, D. R., [1])
(1.5)
UiLfiv)
ViLfiu) =
~ M{Uy Vy U'y V')
dx
when expanded is a formal identity in the variables ( u , v> u', v\ u*, v ff) and x.
We shall prove the following lemma.
Lemma 1.1. In case the system (1.1) is self-adjointt then after an admissible
modification , equations (1.1) will assume a form in which
(1.6) &»/ ^ a /», c%j = bjif dij = dji}
where these functions are of class C 1 in x.
After a suitable modification of (1.1) we will have di9 = da where da is of class
82
SELF-ADJOINT SYSTEMS
[IV]
C1 in x. Upon then equating coefficients of corresponding terms in (1.5) we
find that
(1.7)
From these conditions we see
(1.8)
bij
Pa
— y Hi
an
— fiij)
j
— P}i)
b{j Cij
= <Xijt
Cji - bn
~ OLji}
that
~ a'iy
a%j =
an =
— ocn,
b j i —
an + e
where e*> is a constant of integration. From (1.8) we find that en = 0. For
i > j we now add a constant to bn so chosen as to make en = 0 in (1.8), adding
the same constant to c*,-, as is admissible. From (1.8) we see that = — etj,
so that e%j = 0 without exception. The fourth and fifth conditions in (1.7)
taken with (1.8) now show that
(1.9) Ca s bn.
We have chosen , so as to be of class C 1 while a a and were so given. The
modified coefficients bn will still be of class C1, as will by virtue of (1.9).
The proof of the lemma is now complete.
We are thus led to the following theorem.
Theorem 1.1. A necessary and sufficient condition that the equations (1.1) be
self-adjoint is that after a suitable admissible modification , equations (1.1) take the
form
(1.10) L,(v) = f (Rifl'i + Quod - (Q,Wj + Pifli) = 0
where P,*/, Qa, and Rn are of class C 1 in x and
Rn(x) s Rn(x), Pij{x) s Pji(x).
That the condition of the theorem is necessary ha3 already been proved.
That it is sufficient is readily seen upon taking the bilinear form M as the form
Ui(RijVj H“ Qifl j) (JRijtoj “1“ Qi jU y) .
[2]
SELF-ADJOINT BOUNDARY CONDITIONS
83
Thus in case the equations (1.1) are self-adjoint, they are the Euler equations
of the integral
2 J ' GO y,y')dx
where
(Ml) 211 = RijVWi + 2 Qi}V[vi + PiMi.
We shall term Q a differential form corresponding to the equations (1*1).
A representation of self-adjoint boundary conditions
2. Before defining self-adjoint boundary conditions it will be convenient to
define adjointness relative to a bilinear form.
Let P(u, v) be a bilinear form in m variables (u) and in variables ( v ). Suppose
the matrix of coefficients in P(u , v) is of rank m. Let there be given p homo¬
geneous independent linear forms
(2.0) Ui, • - • , Up (0 < p < m)
in the variables (u) together with m — p homogeneous independent linear forms
(2.1) Vh ... , Vm - p
in the variables ( v ). The conditions
(2.2) I!* =0 (* = 1, * • • , P)
will be said to be adjoint to the conditions
(2.3) V, = 0 (j = 1, • • • , m - p)
relative to the form P(u , v) if P(u , v) vanishes whenever its variables are sub¬
jected to the conditions (2.2) and (2.3).
If the conditions (2.2) are given, a corresponding set of adjoint conditions
can be obtained as follows (B6cher [2]). To the forms (2.0) one adjoins m — p
other forms Up+ 1, • • , Um of such a nature that the forms
(2.4) Uh - }Um
are independent. According to the theory of bilinear forms there will then exist
m independent homogeneous linear forms V { in the variables ( v ) such that
(2.5) P(u, v) S UxVt + • • • + l/.Vt-
The conditions
(2.6) F* = 0
are clearly adjoint to the conditions (2.2).
O' = 1> • • • . m - p)
84
SELF-ADJOINT SYSTEMS
[IV]
Any other conditions (2.3) adjoint to the conditions (2.2) will be shown to be
equivalent to the conditions (2.6) in the sense that a set (v) which satisfies (2.3) will
satisfy (2.6) and vice-versa .
To prove this suppose the conditions (2.3) adjoint to (2.2), and that («;) satisfies
(2.3) . Let V k be one of the forms in (2.6). Choose a set ( u ) such that each of the
m forms J7* in (2.4) is null except the one which multiplies V k in (2.5). For this
choice of ( u ) and ( v ) the form (2.5) must vanish according to our definition of
adjoint conditions. We conclude that V k — 0 for our choice of (v). We have
thereby proved that a set (r) which satisfies (2.3) also satisfies (2.6).
Conversely it now follows from the fact just proved and the fact that the forms
(2.3) and (2.6) are respectively independent, that a set (?;) which satisfies (2.6)
satisfies (2.3).
It will be convenient to represent the conditions (2.2) and (2.3) by means of
linear conditions involving auxiliary parameters. Such sets of conditions will
be regarded as adjoint if as conditions on ( u ) and (t;) they are respectively
equivalent to adjoint conditions of the forms (2.2) and (2.3).
A set of conditions will be termed self-adjoint if equivalent to its adjoint
system.
We return now to a set of self-adjoint differential equations of the form
(2.7) Li(yj) = ^ Ify - ilv. = 0 (i = 1, • • • , n)
where 12 is given by (1.11). We shall assume that the system (2.7) is positive
regular, that is, that
(2.8) Rij(x)wiWj >0 r (t, j = 1, • • ■ , n)
for any set of constants (w) ^ (0), and for x on (a1, a2).
As previously, we set
(2.9) r< = 0,;(V, v'),
regarding this as a transformation from the variables (y, y') to variables (y, f).
We shall also use variables ( f j, ff') and a corresponding set
(2.10) f, = O?j0m »').
Subject to (2.9) and (2.10) the Green's formula takes the form
(2.11) J (y iLi(f\) — Li(rj)fii)dx = — £■»*?» Jal.
We set
(2.12) [i aft - - P(v, f; n, ?)
regarding this as a bilinear form in the two sets of 4 n variables
(v'i, fJ) Oj!', fj) (s = 1, 2;i = 1, ••• ,n).
[2]
SELF-ADJOINT BOUNDARY CONDITIONS
85
To define our boundary conditions we let n and ( be matrices each consisting
of one column, and containing respectively the elements
(2.13) li, vl
(2.14) fi, •** , fi, -f*, , -fn*
Let £ and g be matrices of 2 n columns and p rows, 0 < p < 4n, such that the
matrix 1 1 p, q 1 1 is of rank p. The general boundary problem will now be given the
form
L Q'i ~ 0
(* = 1, ‘
(2.15)
II
*
Conditions (2.15) require that the variables of n and < satisfy p linear, homo¬
geneous, independent equations. By the boundary conditions adjoint to (2.15)
will be meant the conditions adjoint to (2.15) relative to the bilinear form
fj, f). The conditions adjoint to (2.15) then require that the variables
of n and ( satisfy 4n — p linear, homogeneous, independent equations. These
adjoint conditions may be given parametrically as stated in the following lemma.
Lemma 2.1. The conditions adjoint to the conditions (2.15) can be represented
in matrix notation in the form
(2.16) n = q*vy C = P*v ,
where v is a column of p parameters (tq, • • • , vp) and where p* and q* are the ma¬
trices conjugate to p and q.
The conditions (2.16) are equivalent to 4n — p independent linear relations
among the elements of n and (, as one sees upon eliminating the parameters (i>).
To prove that the conditions (2.16) are adjoint to the conditions (2.15) we have
merely to show that the form P(rj, f ; f}, f) =0, subject to (2.15) and (2.16).
But we have the following matrix formula for P, — the addition of an asterisk to a
matrix shall indicate the conjugate of the matrix, —
II P(v,r,n, fill = n*< - C*n.
Subject to (2.16) we find that
||P|| = v*q( - v*fin,
and subject to (2.15) this is seen to be null.
The lemma is thereby proved.
We continue with the following lemma.
Lemma 2.2. In order that the conditions (2.15) be self-adjoint it is necessary and
sufficient that p == 2 n, and that the matrix pq * be symmetric .
It is clearly necessary that p = 2 n.
86 SELF-ADJOINT SYSTEMS [ IV ]
If p = 2 n, a necessary and sufficient condition that the system (2.15) be self-
adjoint is that the matrix equation
pn = q\
be satisfied by all sets ( fj> f) which are given by (2.16). This gives the condition
(2.17) pq* v = qp*v.
Now (2.17) holds for every set ( v ) if and only if pq * is symmetric, and the lemma
is proved.
The following theorem gives a new and basic representation of self-adjoint
boundary conditions. In it there appear just the coefficients which are arbi¬
trary. It gives the second precise link between the theory of self-adjoint bound-
aiy conditions and the theory in the preceding chapters.
Theorem 2.1. Any set of self-adjoint boundary conditions can be given the form
(2.18) ' n - cu = 0,
(2.18) " c* C - bu = 0,
where u is a column of r parameters with 0 ^ r S 2 nf c a matrix of rank r of r
columns and 2 n rows , and b a symmetric matrix of r rows and columns. Con¬
versely any set of conditions of this form is self-adjoint .
We shall first prove that any set of self-adjoint conditions of the form (2.15)
can be given the form (2.18). In the conditions (2.15) suppose q has the rank r
(possibly zero). Without loss of generality we can suppose the conditions (2.15)
are replaced by an equivalent set in which the last 2 n — r rows of q are null.
According to Lemma 2.1 self-adjoint conditions of the form (2.15) are equivalent
to conditions of the form
(2.19) n = q*v, C = P*v,
where v is a column of 2 n parameters. But if the members of the second matrix
equation in (2.19) are multiplied on the left by q and the relation qp* = pq *
used, (2.19) yields the conditions
(2.20) n - q*v} q( = pq*v.
Thus self-adjoint conditions (2.15) lead to conditions (2.20). But the conditions
(2.20) lead back to the conditions (2.15) as one sees upon replacing q*v by n in
(2.20) . Thus conditions (2.20) are equivalent to (2.15) if (2.15) is self-adjoint.
To reduce conditions (2.20) to the required form recall that pq * is symmetric.
Moreover its elements are seen to be null except for a matrix b of elements in its
first r rows and columns. If we let c denote the matrix of elements in the first
r columns of q*f and take ( u ) as the first r of the parameters ( v ), (2.20) takes the
form (2.18) as desired.
Conversely conditions of the form (2.18) are always self-adjoint. To prove
this we note that the elimination of the parameters (u) will yield 2 n linearly
12)
SELF-ADJOINT BOUNDARY CONDITIONS
87
independent linear conditions on the elements of n and To complete the
proof it will be sufficient to show that the bilinear form
(2.21) 1 1 P(v, f) 1 1 = «*{ — C*n
is null, subject to (2.18) and to the corresponding conditions
(2.22) n — cii = 0, c* £ — bii = 0,
where u is a column of r parameters. Subject to these conditions the form
(2.21) becomes
ii*c* C — C *cuf
and upon using (2.22) and (2.18) again, the form finally reduces to
ii*bu — it*b*u s 0.
The proof of the theorem is now complete.
We shall now drop the matrix notation and represent our self-adjoint bound¬
ary conditions (2.18) in the form
(2.23) ' v- ~ c\huh = 0 {s = 1, 2; i = 1, • • • , n),
(2.23) " c2afi - + bhkuk = 0 (h, k = 1, • • • , r),
where || c\h || is a matrix of rank r and || bhk || is symmetric. We shall term
(2.23) ' the accessory end-plane ttt in the space of the 2 n variables v * regarding the
variables ( u ) as parameters. We include the 0-plane 77" = 0 as a special case,
calling it the null end-plane. The symmetric quadratic form bhkuhuk will be
called the accessory end-form. Its value will be regarded as a function of the
point on 7rr represented by (?/).
We see that the general self-adjoint boundary problem with differential
equations in the form
qv, - S2V. = 0 (t = 1, ■ ■ ■ , n),
and boundary conditions in the form (2.23), is uniquely determined by giving an
accessory end-plane 7rr,
(2.24) v9i - c\huh = 0 (s = i, 2; h = 1, • • • , r; 0 g r g 2n)
in w^hich the matrix 1 1 c*ih 1 1 is of rank r, an accessory end-form
(2.25) b hk'Uh'U'k {hj k 1, * > r)
in which the coefficients are symmetric, and a differential form
(2.26) n(v, Vf)
as previously described.
The accessory end-plane and end-form are geometric invariants in the followr-
ing sense.
88
SELF-ADJOINT SYSTEMS
[IV]
A necessary and sufficient condition that two sets of self-adjoint boundary con¬
ditions of the form (2.23) be equivalent is that their accessory end-planes wr consist
of the same points n in the space of the 2 n variables n and that their accessory end-
forms be numerically equal for values of their parameters which determine the same
point on wr-
If two sets of equivalent conditions (2.23)' are given, I say that their accessory
end-planes consist of the same points n. In fact in the space of the 4 n variables
n and £ the given boundary conditions define a 2n-plane 7r2n obviously inde¬
pendent of the parametric representation of the conditions. I say that the
accessory end-plane Tr is the orthogonal projection in the space n, £ of 7r2n on the
coordinate 2n-plane of the variables n. This appears at once from the form of
(2.23). Hence if the boundary conditions are equivalent, there can be but one
end-plane 7rr.
To turn to the accessory end-forms, suppose that we have given a set of con¬
ditions (2.23), and a second and equivalent set of conditions (2.23)' with acces¬
sory end-form
bhkUhUk .
Let (' u ) be any set of r parameters. Corresponding to (u) there exists a unique
point n on irr which satisfies (2.23)', and at least one set £ which then satisfies
(2.23)". Upon multiplying the Mh condition in (2.23)" by uh and summing,
using (2.23)', we find that
(2.27) nifi ~ 77-f* = bhkuhuk.
On the other hand this same set n, £ must satisfy the equivalent conditions
(2.23V with a set (il), and we must have
- v2it* = bhkuhuk.
Thus for sets ( u ) and ( u ) which determine the same point n on 7rr, we have
bhkUhilk = bhkUhUk .
The conditions of the theorem are accordingly necessary.
To prove the conditions sufficient suppose that we have given a non-singular
(in case r > 0) linear transformation
(2.28) uh = ahviiv (A, p = 1, • • • , r)
from parameters ( u ) to parameters (w). Suppose also that an accessory end-
plane 7rr is represented in two ways,
V* = c'ihUhl rj\ = c\hUkj
where
C ip ~ CiHabp
(i = 1, • • • , n; A, p = 1, • • , r),
[3]
BOUNDARY PROBLEMS INVOLVING A PARAMETER
89
and that we have two accessory end-forms such that
bpqUpliq = b hkUh'M'k
subject to (2.28), where
bpq = ahpbhkakq (h, k, p, g = 1, - • • , r).
To prove the conditions of the theorem sufficient we have merely to prove that
the conditions (2.23) are equivalent to the conditions
Vi - c’ipup = 0 (i = 1, • • • , n\ s = 1, 2),
t = t>PQup (p> q = 1, • • • , r).
To prove this statement we first observe that the conditions (2.23) are equiva¬
lent to the conditions obtained by replacing uk by akquqy that is, to the conditions
(2.30)
v\ - KvUp = o,
Cafi ” c)ht2i = bhkakquq.
But if the hth condition in (2.30) is multiplied by ahp and the resulting equations
summed with respect to A, we obtain the conditions
c\Pt\ - = b^u,.
Thus the satisfaction of the conditions (2.23) by variables n, £ entails the satis¬
faction of the conditions (2.29) by the same variables. Similarly the conditions
(2.29) lead to conditions (2.23) . Hence the two sets of conditions are equivalent.
The proof of the theorem is now complete.
It follows from the preceding that we have three numerical invariants asso¬
ciated with each set of self-adjoint boundary conditions, namely, the dimension r
of the accessory end-plane and the index and nullity of its accessory end-form.
These numerical invariants together with similar invariants associated with two
problems are fundamental in what follows.
Boundary problems involving a parameter
3. We shall consider a boundary value problem B involving a parameter a
in such a manner that for each value of <r there is defined a self-adjoint boundary
problem of the sort already defined.
The differential form shall be
(3.1) 2w(tj, i)\ <0 = Pij(x, <r)ViVj + 2Q,J(x, a)v'iVi + #,-;(*, <r)vW,
(i,j = 1 ,•■•,»)
and the accessory end form
(3.2) bhk(o)uhuk (h, k = 1, ■ • • , r; 0 g r ^ 2 n),
90
SELF-ADJOINT SYSTEMS
[IV],
while the accessory end-plane shall be
(3.3) - c\huh = 0 (s = 1,2),
where c\h is independent of a.
We shall consider the functional
I(rj) a) = bhk(a)uhuk + J 2 co(rj, a)dx
evaluated for functions th(x) of class Dl subject to (3.3), and admit problems B
satisfying the five following hypotheses.
A.l. For any real number a and for x on the interval (a1, a2) the functions
(3.4)
B Qij) Bij
dRa dQa
dx ’ dx 9
bf,k
shall be continuous.
A. 2. The matrix of elements c*h shall be of rank r, and the matrix of elements
bhk symmetric.
A. 3. For each value of a and of x on (a1, a2) and set (w) ^ (0),
/?.,-(*, (r)wiWj > 0.
A, 4. For —a sufficiently large the functional I(rj, a) shall be positive definite.
A. 5. The accessory form shall decrease monotonically as a increases. It may
in particular be independent of a. For (77) ^ (0) and x fixed, c 0(77, tj', a) shall
definitely decrease as a increases.
By a characteristic solution of B is meant a set of functions r)t (x) of class C2
in x which with constants ( u ) and a satisfies the boundary problem By but which
i:: not identically null. The corresponding constant a is called a characteristic
root. By the index of a characteristic root a is meant the number of independent
characteristic solutions corresponding to that root. In counting characteristic
roots each root will always be taken a number of times equal to its index.
Corresponding to the functional
/*«2
Ja = %bhk(o-)othak + / u(y, ?/, a)dx
subject to the end conditions
Vi - Cih<*h = 0,
for each value of a we can set up a special index form Q(z, a), just as the special
index form Q(z , X) was set up at the end of §1, Ch. Ill, for each value of X. We
shall then have
Q(z, a) = bhk(a)uhuk + J* 2oj(t7, 77', a)dx ,
[3]
BOUNDARY PROBLEMS INVOLVING A PARAMETER
91
where (ij) is the secondary broken extremal E “determined” in B by the set (z)
for the given cr. More explicitly if we set
(z) = (mi, • • • , ur, z\, ■ ■ ■ , z\, ■ ■ ■ , z\, • ■ • , zl),
the end points of E are given by the conditions
(3.5) 77J - c*huh = 0, x = aa,
and the intermediate corners or vertices by the conditions
Vi(aQ) = z? (q = 1, - • • , p)
where x — aq is the ?th “intermediate” n-plane.
Essentially as in §2, Ch. Ill, we can prove the following theorem.
Theorem 3.1. The form Q(z , a) is singular if and only if a is a characteristic
root. If a is a characteristic rooty the nullity of Q(z, cr) equals the index of a, and
the index of Q(zy a) equals the number of characteristic roots less than cr.
In reviewing the proof in Ch. Ill we call attention to the fact that the right
member of (2.8) will here be replaced by the expression
[bhk(cr') — bhk(a")\uhuk + J [2c o(rj, rj'} a') — , 77', a")]dx.
Moreover for o" < a' and (77) ^ (0) this functional is negative, as follows from
Hypothesis A. 5. One continues as before. We recall that Hypothesis A. 4
affirms that for — cr sufficiently large I(rj, a) is positive definite. This was proved
in the earlier case, but is an hypothesis here.
As a corollary of the theorem we see that the number of characteristic roots
less than a given constant is finite, and that accordingly the characteristic roots
are isolated.
We shall arrange the characteristic roots in a sequence
(3.6) (Jo ~ cr 1 ~ (72 — <r3 ^
in which each root appears a number of times equal to its index. The number
of roots may be either finite or infinite as examples will show.
We shall say that the problem B depends continuously on a parameter a for
a near a0 if the functions in (3.4) are continuous in x, a, and a , and if the remain¬
ing Hypotheses A are satisfied for each value of a.
We shall prove the following theorem.
Theorem 3.2. If the kth characteristic root in the problem B exists for a = a0,
it exists and varies continuously with a for a sufficiently near a0.
For a — ao let <rr and a" be two constants respectively less and greater than
<Tky so chosen as to separate c tu from the characteristic roots not equal to 0*.
Designate the roots equal to cr* when a = a0, including ck, by
(3.7) *1 ■■■ ,ol + v.
92
SELF-ADJOINT SYSTEMS
[IV]
The forms Q(z} a') and Q(z , a ") set up for a = a0 will be non-singular and possess
indices respectively equal to h and h + v + 1.
If now the parameters a be continuously varied, the coefficients in these forms
will vary continuously. For a sufficiently small variation of a they will remain
non-singular and hence unchanged in index. After such a variation there will
then still be v + 1 roots ohf • • • , <rh+v between <r' and <r". Since crf and <r"
can be initially taken as near as we please to a°k we see that ak will vary continu¬
ously with a as stated.
The theorem is thereby proved.
Comparison of problems with different boundary conditions
4. We shall now compare two problems B and B i with a differential form
01(77, t)', a) in common. The accessory end-planes in B and B x will be denoted
respectively by 7rr and irri where r and rx are the dimensions of these end-planes.
Let 7r r and 7rri be respectively represented by means of parameters (u) and (V).
Let the corresponding accessory end-forms be
(4.0)
bhk{o)V’hUk
(h, k= 1, •
• • , r)
and
(4.1)
KM)Ku\
(P, Q = 1, • ‘
• > >'i)-
If 7 rr and 7 rri are identical and identically represented with ( u ) = ( ul ), we
shall call
(4.2) d(u, a) = [blhk(<r) - bhk(a)]uhuk
the difference form corresponding to Bi and B.
If on the other hand ttTx is a section of ttT) and if
bXpq(<r)ulpu\ = bhk((r)uhUk
when (u1) and ( u ) determine the same point on 7rri, then Bx will be called a sub¬
problem of B.
We shall now give three comparison theorems, one of each of the following
types:
I. A comparison of a problem with a sub-problem.
II. A comparison of two problems with a common accessory end-plane.
HI. A comparison of two general problems.
In all three cases we suppose the problems have a common differential form.
Theorem 4.1. Let there be given a problem B and sub-problem Bx with accessory
end-planes irr and irri respectively . If v and vx are respectively the numbers of
characteristic roots in B and Bx less than a given constant a, then
(4.3) i> — (r — ri) ^ rg v.
To prove the theorem we refer 7rr to parameters (u) in such a manner that
Tri is obtainable from 7rr by setting the last r — rx of the parameters (u) equal to
[ 4 ] PROBLEMS WITH DIFFERENT BOUNDARY CONDITIONS 93
zero. For this choice of end parameters let Q(z , a) and Qi(z, a) be the special
index forms corresponding respectively to B and Bh using the same intermediate
ft- planes. The form Q^z, a) can be obtained from Q(z, a) upon setting
«r, -f- 1 = * * * = 2r = 0
and renumbering the remaining variables. According to Lemma 7.2 of Oh. Ill
the index of Qi(z, a) must then lie between v and v — (r — rj) inclusive.
The theorem is accordingly proved.
We shall prove the following corollary.
Corollary. The number of roots in B on any open or closed , finite interval of
the a axis differs from the corresponding ti umber for a sub-problem By by at most
r — ru
Let vf and v" be respectively the numbers of eharact eristic roots less than a'
and a" in B with o' < o" . Let v, and vx be the corresponding numbers for Bu
By virtue of (4.3) there exist integers m' and m " such that
vf — v[ + m* y 0 g m' ?> r — r\,
v” = v\ + m”, 0 g m” g /• - rh
so that
v" — == vl — v[ + m" — m'f | m" — tnf | Sr — rx.
But v" — v' and v[ ~ v[ are respectively the numbers of roots in B and Bx on
the interval
(4.4) o' ^ a < o" ,
so that the corollary is proved for intervals of the type (4.4).
Now corresponding to any finite interval whatsoever there exists a closely
approximating interval of the form (4.4) containing the same roots of B and B} .
The corollary is accordingly true in general.
We term a problem with end conditions rj \ — 0 a problem with null end-plane
or null end points. Every problem B possesses a sub-problem with null end
points. Of all sub-problems of B the problem with null end points possesses
the minimum number of roots less than a constant a*. In a problem with r end
parameters there will be at most r more roots less than <x* than appear in t he
corresponding problem with null end points.
Our second theorem is the following:
Theorem 4.2. Suppose B i and B have a common accessory r-plane and common
differential form. Let d(u, a) be a corresponding difference form (4.2), and let
N(<r) and P(cr) be respectively the indices of d(u , a) and ~~d{u, a). If v(a) and
V\{<j) are respectively the numbers of characteristic roots less than a in B and B^thcn
(4.5)
v(<r) — P(er) ^ 2u(<r) ^ t’(<r) + N((t).
94
SELF-ADJOINT SYSTEMS
[IV]
We suppose the end conditions in B and B i represented in terms of common
parameters (u) and let Q(z, a) and Qfz , a) then be the corresponding special
index forms set up with common intermediate n-planes. We have
Qi(z, or) — Q(zf a) ss d(u, cr)
where the first r of the z s are given by (u). It then follows from Lemma 7.3
of Ch. Ill that (4.5) holds as stated.
Suppose the accessory end-planes 7rr and 7rri of the two problems B and Bx
intersect in a £-plane irt. Let B[ and Bl be respectively the sub-problems of B\
and B for which t t is the accessory end-plane. Let 6(w, a) be a corresponding
difference form for B{ and B l. The form 8(u, a) will be called a maximal differ¬
ence form for B i and B. We denote the index of 8(u, a) by N(a) and that of
-8(u} a) by P(ct).
Our third theorem can now be stated as follows;
Theorem 4.3. Let toe re be given two problems By and B with common differential
form y and with accessory end-planes i rr and tcTx intersecting in a t-plane irt. Let
5(u, a) be a corresponding maximal difference form with its indices N(a) and P(a).
If v(a ) and V\{a) arc respectively the numbers of roots of B and Bx less than the
constant a, then
(4.6) v(a) — P(<x) — r + t g l’3(a) ^ v(a) + N(a) + r i — t.
Let h i(o-) and h(a) be respectively the numbers of characteristic roots less thar
a in Bl and Bl. According to Theorem 4.2 there exists an integer q(cr) such that
(4.7) hfa) = h(a) + q(c), ~P(a) ^ q(a) ^ N(a ).
But according to Theorem 4.1 we have
Vi (<x) = hi(a) + dh 0 < dx S rt - t,
(4.8)
v(cr) = h(a) + d, 0 g d ^ r — t.
From (4.7) and (4.8) we find that
Vi (<t) - v(a) = q(c t) + di — d,
from which we see that
v(a) + q{<r) — d g Vi(<r) ^ v{c) + q(a) + d\.
The required inequalities now follow from the inequalities in (4.7) and (4.8)
limiting q, d} and d\.
Theorem 4.3 reduces to Theorem 4.1 when rx = t and P = N — 0. It reduces
to Theorem 4.2 when r = rx = t.
Corollary. The number of characteristic roots on any finite interval of the a
axis differs by at most 2 n for any two problems with common differential form and
with end conditions which are independent of a.
15]
A GENERAL OSCILLATION THEOREM
95
To prove this corollary it will be sufficient to prove that the extreme members
of the inequalities (4.6) differ by at most 2 n.
This difference is
{4.9) N + P + r + ri — 2t^r-\-ri — t
since N + P ^ t. But since 7rt is the intersection of 7rr and Tri we must have
r + ri — 2n g t
or
r + ri — t S 2 n.
The corollary now follows from (4.9).
A general oscillation theorem
5. Let A(x, a) be an n-square determinant of elements 7/,;(.r, a) whose columns
are solutions of the Jacobi equations such that
TUi(a\ a) = 0, v'i,(a\ <r) = 8] (i, j = 1, • • • , n).
Recall that A(x, a) vanishes at each conjugate point x = c of x = a1 to an order
equal to the index of x = c as a conjugate point. A zero x — c of A(x, a) will be
counted a number of times equal to its index.
We have already seen that A(x, a) vanishes on the interval a1 < x < a2 a
number of times exactly equal to the number of characteristic roots less than a
in the problem with null end points. This is a first oscillation theorem. But
by virtue of Theorem 4.1 we can compare the number v(<r) of characteristic roots
less than a in any problem B with the number less than a in the corresponding
null end point problem. We thereby obtain the following general oscillation
theorem.
Theorem 5.1. If r is the dimension of the accessory end-plane of a problem B
and v(<r) is the number of characteristic roots in B less than cr, then A(x, a) vanishes
on the interval a1 < x < a2 at least v(a) — r times and at most v(a) times.
The case n == 1 has been extensively treated by various mathematicians.
See Ince [1], p. 247. In spite of this fact it is possible to use the preceding
theorem to obtain narrower limits on the number of zeros of characteristic solu¬
tions than have been obtained before.
In case n = 1 we first note that the only possible values of r, the dimension of
the accessory end-plane, are r = 0, 1 or 2. The determinant A(x, a) reduces to a
solution y(x) of the Jacobi equations for which
7](al) = 0, n'W) = 1.
Let a* be a characteristic root in a problem B. The root < r* can be simple or
double. We suppose a* represented by crp in case it is simple, and by <rp =
c p— -l in case it is double. In either case we understand that ap is the (p + l)st
96
SELF-ADJOINT SYSTEMS
[tvi
root, counting roots according to their indices. Let r represent the integer 1 or 0
according as x = a2 is or is not a conjugate point of a1. We shall prove that
A(xf a*) vanishes at least p times on the interval
(5.1) a1 < x < a2,
where
(5.2) n = p+ l — r-r.
Let B° be the boundary problem with null end points. Let cr° be a constant
which separates a* from any characteristic root in B or B° for which <j >
According to the preceding theorem A(x, a0) vanishes at least
v(a°) — r = p + l — r
times on the interval (5.1). Hence A(xf a*) vanishes at least p + 1 — r — r
times on (5.1) as one sees upon continuously decreasing a from cr° to a*.
We can now establish the following theorem.
I. In case n = 1, a characteristic solution rj* corresponding to a simple root
vanishes at least p — 1 times and at most p + 1 times on (5.1). A characteristic
solution rj* corresponding to a double root ap — vanishes either p — 1 or p
times on (5.1).
Observe that 77* vanishes on (5.1) at least as many times as A(x, a), that is at
least p + 1 ~ r — r times. If r = 0 or 1, or if r — 2 and r = 0, this is at least
p — 1 times.
There remains the case r = 2, r = 1. In this case we note that 77* cannot
vanish at a1 and a2, because we see then that the parameters (w) in the boundary
conditions would be null, and hence f1 = f2 = 0 for 77*. We would then have
77* e 0 which is impossible. Nor can 77* vanish at a1 or a2 alone, since the
hypothesis that r — 1 would then make 77* vanish at both a1 and a2. The
solution 77* must then vanish on (5.1) once more than A(x, a*), or at least p — 1
times as stated.
On the other hand A(x, a*) vanishes at most p times on (5.1) in case av is a
simple root, as follows from Theorem 5.1. Hence in case <rp is a simple root 77*
vanishes at most p + 1 times on (5.1). A similar use of the theorem in case
<rp = 1 is a double root shows that 77* vanishes at most p times on (5.1).
Thus statement I is proved.
Note that the spread between p — 1 and p + 1 is 3. Compare Ettlinger
(1.2) where the difference between the limits is 5.
Bocher [2] has treated the periodic case at length. Ince [1] has summarized
Bdcher’s results in a theorem on p. 247, loc. cit. In this theorem the existence
of infinitely many positive characteristic roots is affirmed. This result will
follow from our general existence theorem to be established in a later section.
The theorem of B6cher also asserts that each characteristic solution vanishes an
[6]
THE EXISTENCE OF CHARACTERISTIC ROOTS
97
even number of times. This follows from the periodicity of the boundary condi¬
tions. The principal part of Bbcher’s theorem is a special case of the following:
II. In case n = r = 1, a characteristic solution corresponding to a simple root
op vanishes either p or p + 1 times on the interval a1 g x < a2, ivhile a characteristic
solution corresponding to a double root op — op~i vanishes exactly p times on
a1 ^ x < a2.
To prove this statement recall that A(x, a*) and hence n* will vanish at least
p + 1 — r — t times on (5.1). In case r = 1 and t = 0, this is at least p times.
Hence in this case y* vanishes at least p times on (5.1). In case r = t — 1,
A(x, a*) vanishes at least p — 1 times on (5.1) and at least p + 1 times on
a1 ^ x ^ a2. Hence in this case rj* vanishes at least p times on a1 S x < a2.
The remaining facts in II follow from I in case rj* does not vanish at a1. In
case rj* vanishes at a1 its zeros are those of A(xf o*). According to the relations
between conjugate points and characteristic roots in the problem with null end
points we see that in this case rj* vanishes exactly p times on (5.1) in case o *
is a simple root, and exactly p — 1 times in case o* is a double root. Hence in
case rj* vanishes at a1 it vanishes p + 1 times on a1 g x < a2 if o* is a simple
root and p times if o* is a double root.
The proof of II is complete.
The existence of characteristic roots
6. We begin with the following lemma.
Lemma 6.1. If the x-coordinate xk(o) of the kth conjugate point of x — a1 exists
for o = o*} it exists and decreases as o increases from o* neighboring a*.
Let Xk(o*) = a". Let the index form Q(z) o) be set up for the problem with
null end points at x — a1 and x = a" , and with a near o*. The index plus the
nullity of Q(z, >o*) will be at least k. Accordingly Q(z, a*) will be non-positive
on a fc-plane r* through the origin in the space ( z ). But
Q(z, a) = J ^ 2u>(t?, V, o)dx,
where ( rj ) represents the broken secondary extremal determined by (z).
Suppose now that o is slightly larger than a*. Recall that ^(t?,^' ,<r) is
assumed to be a decreasing function of o for (??) ^ (0). We see then that for
o > <7*, Q(z, o) < 0 on Tk, and that accordingly the index of Q(z, o) will be at
least k. Hence the fcth conjugate point of x = a1 must precede a" for o > o*y
and the lemma is proved.
Let the segment of the x axis
ax ^ x ^ a2
be denoted by y. We state the following theorem .
98
SELF-ADJOINT SYSTEMS
[IV]
Theorem 6.1. A necessary and sufficient condition that there exist an infinite
number of characteristic roots in an admissible boundary problem B is that there
exist a point P on y with the following property. Corresponding to any segement y
of y which gives a neighborhood of P on y there exists an admissible curve X which
joins P to a point Q 9^ P on 7, along which (77) ^ (0) and on which
(6.1)
L
(x)(r), rj' y <j)dx ^ 0
for all values of a exceeding a sufficiently large positive constant 07.
Recall that any two points on the x axis which are conjugate can be joined
by a secondary extremal L along which (77) ^ (0) and (6.1) holds.
Suppose the condition of the theorem were not necessary. It would then
follow from the preceding lemma that with each point P on y there could be
associated an open interval I which contained P in its interior (or on its bound¬
ary if P is an end point of 7) and which contained no conjugate point of Py no
matter how large a might be. The whole segment 7 could then be covered by a
finite set of such intervals /. But according to a separation theorem to be
established in §8 there can be at most n conjugate points of x — a1 on each open
interval /, and hence at most a finite number N of conjugate points of x — a1 on
7, where N is independent of a.
But this is impossible. For we are assuming that there are infinitely many
characteristic roots a in the problem By so that there will necessarily be infinitely
many characteristic roots in the problem with null end points. For a sufficiently
large there must then be arbitrarily many conjugate points of x = a1 on 7, in
particular there must be more than N such conjugate points. The condition
of the theorem is accordingly necessary.
We shall now prove the condition sufficient.
Suppose the condition of the theorem is satisfied by a point P. One sees that
one at least of the two sides of P on 7 must have the property that the condition
of the theorem is satisfied by curves X whose end points Q all lie on that side of P.
Suppose the side preceding P has this property.
For <7 sufficiently large, say a > or1, the first conjugate point xi(cr) of a1 follow¬
ing a1 cannot follow P. For otherwise for a > crl there could be no conjugate
point of P between P and x = a1 as follows from our separation theorem,
Theorem 8.3. In such a case (6.1) could not be satisfied for Q between x = a1
and P.
As a increases, Xi(cr) will decrease in accordance with the preceding lemma.
For <r > <rl and sufficiently large, say a > a2} the second conjugate point x2(<r)
of a1 cannot follow P for the reasons cited in the case of Xi(a). Reasoning thus,
one sees that in general for a sufficiently large there must be arbitrarily many
conjugate points of x = a1 preceding P. It follows from the oscillation theorem
of the preceding section that there must be infinitely many characteristic roots
in B.
[7]
PROBLEMS POSSESSING DIFFERENT FORMS co
99
The condition of the theorem is accordingly sufficient and the proof is complete.
We note the following corollary of the theorem.
Corollary. A sufficient condition that there be infinitely many positive char¬
acteristic roots in B is that Ci}(r}} rj' } a) involve a only in terms of the form
(6.2) — a a-ifix) rjiVj
and that the form (6.2) be positive definite for each x on (a1, a2) and for a < 0.
To show the power of the preceding methods we shall briefly indicate an
important extension of this corollary.
If Hypotheses A.l, A. 2 and A. 3 are satisfied , but the form (6.2) is assumed
positive for a < 0 merely for one point x0 and one set (r/°) ^ (0), there still exist
infinitely many characteristic roots greater than any constant a*.
The special index form Q(z, <y) can still be set up for each a as before. For
any finite range of values of a the same set of intermediate n-planes can be used.
But here the number of characteristic roots 0 on a given interval <r' ^ a < o "
may very well be infinite
As in the earlier case the nullity of the form Q(z, a) equals the index p of o
as a characteristic root, and in case a* is an isolated root the index of Q(z, a)
can change by at most p as a passes through a*. But in the present case, a
priori at least, the index may increase, or decrease, or remain constant . Let
v” be the index of Q(z, a") and v' the index of Q(z , a'). If 6 is finite, then upon
varying o from o' to a" inclusive, we see t hat
(6.3) <9 £ I v" - v9 |.
Previously (6.3) was an equality. As it stands (6.3) can still be used to prove the
existence of infinitely many characteristic root s.
Let I be an interval which contains x0 in its interior and is so small that for
the given (770),
au(: r) uX >0
on 7. One sees readily that there must be a conjugate point of x = a1 on each
of any set of distinct subintervals of 7, for a sufficiently large and positive
(Morse [16], p. 543). But as previously and with no alteration in the proof, the
index of Q(z, a) will differ by at most r from the number of conjugate points of
x = a1 on a1 < x < a2 for the given a. If for a given a, <r" is chosen sufficiently
large, the change in index of Q(z, <r) as a increases from a' to a" must be arbi¬
trarily great, so that from (6.3) we see that 0 must be arbitrarilv great.
There must then be arbitrarily many characteristic roots greater than a',
and the statement in italics is proved .
With this digression we return to Hypotheses A.
Compaxison of problems possessing different forms a>
7. In this section we shall consider two problems B and Bf satisfying Hypoth¬
eses A and possessing a common accessory end-plane wr. We suppose further
100
SELF-ADJOINT SYSTEMS
[IV]
that B and B' possess infinitely many characteristic roots , If one wishes one
can drop this last assumption, adding the qualification “if <?h exists” to each
statement about <rh.
We suppose the common accessory end-plane is represented in terms of the
same parameters (u) in both problems. The accessory end-form of B ' minus
that of B will be denoted by
(7.0) Abhk(<r)UhUk {h, k = 1, • • • , r)
and the differential form of Bf minus that of B as given in §3 will be denoted by
(7.1) Ao>(t7, t]', <r).
The ( k + l)st characteristic roots of B and B' will be respectively denoted by
crk and ak. By the difference problem D corresponding to B' and B in the order
written, will be understood the problem in which the differential form is the
form (7.1), the accessory end-plane is the end-plane 7rr, and the accessory end-
form is the form (7.0).
The case in which Aw s= 0 has been treated in part in §5. We shall here
consider the case where the difference problem satisfies Hypotheses A, and prove
the following theorem.
Theorem 7.1. If the difference problem D corresponding to problems B' and B
satisfies Hypotheses A and possesses ph characteristic roots less than orhy then
(7.2) (h = 0,1, .■•).
Let the special index forms corresponding to B', Bf and Z>, set up with the
same intermediate n-planes, be denoted by Q ', Q} and Q°, respectively. We
have
[a>
2w(r),r),}cr)dx + Abhk(a)uhUk'i- I 2Aw (77, 77', a)dx
where (77) is determined by ( z ) in the problem B'. From (7.3) we see that
(7.4) Q'(z, a) ^ Q(z , a) + Q°(z, a).
Let k * and k be respectively the numbers of roots less than ah in B' and B . It
follows from (7.4) that for a = <rh the index of the form on the right of (7.4) is
at least k But it follows from Lemma 7.3 of Ch. Ill that the index of a sum of
two quadratic forms is at most the sum of the indices of the two forms. Thus
(7.3) Q (Zy(r) — bhk(KG')V'hW>k
:
(7.5)
If (7.2) were false and
k' g k + ph.
[7]
PROBLEMS POSSESSING DIFFERENT FORMS co
101
we would have
*' ^ h + ph + 1, k ^ h
leading to a contradiction of (7.5). Thus (7.2) holds as written.
(a). We shall now prove that the equality can hold in (7.2) for a given h only if
Bf B'y and D have in common at least one characteristic solution with root a = ah.
We suppose then that
(Jh+ph = ah
for a given h. For this h there will exist an {h + ph + l)^plane ? r' through the
origin in the space ( z ) such that
(7.6) Q'(z, <fh) ^ 0 (on 7r'),
since the index plus the nullity of the form (7.6) is at least h + ph + 1. If ^ is
the number of the variables ( z ), there will exist a (/* — //) -plane tt through the
origin such that
(7.7) Q(z, trh) ^ 0 (on tt).
Furthermore i r' and 7r can and will be so chosen that on them the forms in (7.6)
and (7.7) respectively will be zero only if (z) is a critical point of these forms.
Now 7r ' and tt will intersect in a hyperplane 7r° of dimensionality at least
( h + ph + 1) + (m — h ) — /x = ph + 1.
From (7.4) we see that
(7.8) Q°(z, ah) ^ 0 (on t r°).
But since the form (7.8) has the index ph there must be a straight line X on t r°
on which the form (7.8) vanishes. A comparison of (7.4), (7.6) and (7.7) shows
that the forms (7.6) and (7.7) also vanish on X. Hence each point of X must be a
critical point of the forms (7.6) and (7.7).
We see then that the curve *17° (x) determined in B' for <r = crn by a point
(z°) ^ (0)
on X, will represent a characteristic solution in B'. For such a point (7.4)
will be an equality. But turning to (7.3) we see that (7.4) can be an equality
only if the curve rjlix) is a secondary extremal in B and D as well as in B
Since ( z° ) is a critical point of the forms (7.6) and (7.8), the curve ^(x) must
be a characteristic solution in B and D as well as Bf.
The statement in italics is thereby proven.
We have the following corollary.
Corollary. For the condition
(7.9)
<r'h § o-*
102
SELF-ADJOINT SYSTEMS
[IV]
to hold it is sufficient that the difference problem, I) satisfy Hypotheses A and possess
no characteristic root less than a^
With the aid of (7.3) one immediately verifies the truth of the following
generalization of the Sturm-Liouville comparison theorems.
In order that (7.9) hold it is sufficient that
(7.10) Abhk(a)uhUk + J 2Au(rt, v', <r)dx ^ 0
for all curves (rj) of class Dl and sets (u) which satisfy the secondary end conditions
with (77). In order that a'h > ah it is sufficient to exclude the equality in (7.10) for
to * (0).
Boundary conditions at one end alone
8. We return now to the differential form 0(17, ff) of §1 and corresponding
Euler equations
(8.1) = 0 (*=!,••■, rt)
not involving a parameter a assuming that Rij(x)WiWj > 0 for {w) 9^ 0. Our
boundary conditions shall be conditions at x = a1 of the form
(8.2) pav) = g a?) (i, j = 1, • • • , v),
where the coefficients pa and q a are constants and the matrix || pa, qa || has
the rank n. We say that the conditions (8.2) are self-adjoint at x = a1 if subject
to (8.2) and to the conditions
(8.2) ' pijff) = qal)y
the bilinear form
(8.3) v)i\ - f Wi = 0.
We assume that the conditions (8.2) are self-adjoint.
We see that a necessary and sufficient condition that the conditions (8.2) be
self-adjoint at x = al is that conditions (8.2) together with the n conditions
rj * — 0 form a system self-adjoint in the sense of §2. It follows from the results
of §2 that the conditions (8.2) can be given the form
(8.4) ' ? u — cihuh = 0 (i = 1, ■ ■ • , n; A = 1, • • • , r; 0 r <£ w),
(8.4) " cih{\ - bhkuk = 0 (hf Jc - 1, • • • , r),
where 1 1 Ca 1 1 is a matrix of constants of rank r and 1 1 bkk 1 1 a symmetric ma¬
trix of constants.
We shall give two additional interpretations of the conditions (8.2) of which
the first is in terms of transversality.
[8]
BOUNDARY CONDITIONS AT ONE END ALONE
103
The end conditions (8.4)' require that the initial point (xy 77) lie on an r-plane
Lr. Conditions (8.4) applied to an extremal satisfying (8.1) require that this
extremal cut Lr transversally relative to the functional
[* 1
(8.5) J = \bhkUhUk + I 12(t?, i ]')dx.
We take this statement as a convention when r = 0.
We can give a second interpretation of the conditions (8.2) in terms of the
conjugate families of von Escherich defined in §3, Ch. III. We begin by choos¬
ing a base b of n independent sets (771, f!) which satisfy (8.4) with parameters
(u), and upon which all other sets which satisfy (8.4) are linearly dependent.
Let || rji j(x) II be an n-square matrix whose columns represent the extremals
(77) determined at x — a] by the members of the base b. The family of extremals
cutting Lr transversally relative to (8.5) can be represented in the form
(8.6) rjt(x ) = 7 uj(x)vj (iy j = 1, • • • , n)
where (v) is a set of n constants which serve as parameters of the family. I say
that any two members of the family (8.6) satisfy the identity
(8.7) rji (x)£i(x) - ti(x)f}i(x) S 0 (i = 1, • • • , n).
In fact the left member of (8.7) is known to be identically constant, and this
constant must be zero since the conditions (8.2) and (8.2)' entail the satisfaction
of (8.3). Thus the conditions (8.4) define a family of extremals conjugate in the
sense of von Escherich.
Conversely the members of any conjugate family F satisfy conditions of the
form (8.4). For the initial values (771, f1) of members of F depend linearly upon
the initial values of members of the base used to define the family F . We see
then that these initial values (77*, fl) of members of F must satisfy n independent
linear conditions L. Now (8.7) will be satisfied identically by any two members
of F and in particular will be satisfied at x = a1. The conditions L are then self-
adjoint at x — a1 in accordance with our definition of self-ad jointness at x = a1,
and can accordingly be put in the form (8.4).
We summarize in the following theorem.
Theorem 8.1. The conditions (8.4) have the following three interpretations.
I. They have the form of the most general boundary conditions which are self-adjoint
at x = a1. II. They define the general conjugate family of extremals of the differ¬
ential equations (8.1). III. They define the n-parameter family of extremals which
are cut transversally by the r-plane (8.4)' relative to the functional (8.5).
Recall that the focal points of the conjugate family F are defined as the points
x = c at which the determinant | 77 i?(x) | of a base of the family vanishes. Each
focal point x — c of F is assigned an index equal to the number of independent
solutions of the family which vanish at x = c, and each focal point is counted a
number of times equal to its index. If a focal point x — c has the index n, then
104
SELF-ADJOINT SYSTEMS
[IV]
the focal points of F other than x — c may be regarded as the conjugate points
of x = c.
We shall say that the given boundary problem depends continuously on a
parameter yf if the coefficients in Q(rj, rj') together with the derivatives R \i and
Q'ij are continuous in x and yt while the coefficients cih and bhk are continuous in
ijl. We suppose that the matrix 1 1 cih 1 1 remains constantly of rank r and the
matrix 1 1 6^* 1 1 remains symmetric. Let the boundary problem thereby
defined be denoted by We suppose that y is confined to values near /x = 0.
We shall prove the following theorem.
Theorem 8.2. If the kth focal point following x = a1 of the conjugate family F
defined by exists for y = 0, it exists and varies continuously with yfor y sufficiently
near 0.
Let x — c be the fcth focal point of F following x = a1 when y = 0. Let x — cl
and x — c" be two points on the x axis separating x — c from the other focal
points of F . We suppose c' < c". To the end conditions of the form (8.4)'
in we adjoin the conditions rj] =0, obtaining t hus the conditions
rj) - cih(y)uh = 0 (i = 1, • • • , n; h = 1, • ■ , r),
Corresponding to these end conditions and to the functional J in (8.5), here
depending on y, we set up the special index form of §1, Oh. III. We take a 2
successively as c' and c", and let the corresponding index forms be denoted by
H'(z, y) and H"(z, y).
The forms II' (z, 0) and //"(z, 0) are non-singular since c' and c" are not focal
points of a1. For a sufficiently small variation of y they will remain non-singular
and their indices remain constant. But their indices are respectively the
numbers of focal points preceding c' and c" and following x = a1. Thus the kih
focal point of x = a1 must lie between x = c' and x = c" for y sufficiently near
y — 0. The theorem follows from the fact that c' and c" can be taken arbitrarily
close to x — c.
We shall now establish a theorem on the interrelations of the focal points of
any two conjugate families (Morse [10]).
Theorem 8.3. If two conjugate families F and F° have p linearly independent
solutions in common , then the number of focal points of F on any interval y ( open or
closed) differs from the corresponding number for F° by at most n — p.
Let a1 and a2 be so chosen that a1 is not a focal point of F or F°f and that the
interval a1 < x < a2 includes just the focal points of F and F° on y . Since
x = a1 is not a focal point of Fy at a1 the members of F will satisfy conditions of
the form
(8.9)
v\ = M,
S\ =
(i, j = 1, • • • , n),
[8]
BOUNDARY CONDITIONS AT ONE END ALONE
105
where = {*,•<. Similarly at a 1 the members of F° will satisfy conditions of the
form
(8.10) v\ = w„ fi =
where The quadratic form
(8.11) A(w) = (f?,- - ix,)uiU, (i,j = 1, • • • , n)
will be called the difference form at x = a1 corresponding to F° and jF. Now a
necessary and sufficient condition that the members of F and F° determined by a
set ( u ) be identical is that
(fii - fwK = 0 (j, j = 1, • ■ ■ , n).
We see then that the number of independent solutions common to F and F°
equals the nullity of the difference form A(w).
We nowr adjoin the condition rj \ = 0 to the conditions (8.9) and (8.10) thereby
obtaining two new boundary problems B and B° . Corresponding to B and B°
and the functional (8.5) we set up the special index forms Q(z) and Q°(z) re¬
spectively as in Ch. Ill (X = 0). We use the same intermediate n-planes in
B and B°. We then have
(8.12) Q\z) - Q(z) = A (u)
where the variables (u) equal the first n of the variables (z). It follows from
Lemma 7.2 of Ch. Ill that if v and r° are respectively the indices of Q and Q°,
and N and P are respectively the indices of A (u) and — A (u), then
V - P g v° g V + N.
Hence
| v° — v | g P + N == n — p.
The theorem now follows from the fact that the indices of Q and Q° are re¬
spectively the numbers of focal points of F and F° on a1 < x < a2.
The two preceding theorems enable us to prove the following :
Theorem 8.4. The kth conjugate point of a point x = cfollouring x = c advances
or regresses continuously with x ~ c as long as it remains on the interval on which
the problem is defined.
Choose x = a1 as a point preceding x = c and not conjugate to x = c. The
conjugate family Fc consisting of the solutions of (8.5) which vanish at x = c,
will satisfy conditions of the form
v\ = Ui (i,j = 1, • • ■ > n),
(f«j = ("/•)>
= iott,
106
SELF-ADJOINT SYSTEMS
[IV]
at ct where the coefficients f xj will be continuous in c at least for a sufficiently
small variation of c. That the Mh conjugate point xk (c) of x = c varies con¬
tinuously with x = c now follows from Theorem 8.2.
Suppose next that c increases from c0. Then xk(c) £*(c0), at least after a
sufficiently small increase of c from c0. For otherwise there would be infinitely
many conjugate points of xk(co) near x = c0. Now there are at most k — 1
focal points of FCq on the interval
(8.13) Co < x < xk(c0).
If xk(c) decreased as c increased, at least n + k focal points of Fc would thereby
appear on the interval (8.13) contrary to Theorem 8.2. Hence xk(c) increases
with c . It follows that xk (c) must decrease as r decreases and the theorem is
proved.
We shall give a typical comparison theorem.
Theorem 8.5. Let F and F° be two conjugate families for which x = a1 is noi a
focal point. If the difference form
A(w) = - Ui )UiUj (i, j = 1, ■ , n)
of (8.11) is positive definite , the kth focal point of F° following x - a1 will be pre¬
ceded by the kth focal point of F.
We use the notation of the proof of Theorem 8.3 taking a2 as the kth focal
point of F°. We are led to the relation
(8.14) Q\z) - A(u) = Q(z)
of (8.12).
If there are h focal points of F° on the interval a1 < x S a2, Q°(z) will be nega¬
tive semi-definite on an A-plane irh through the origin. Moreover irh can be so
chosen that Q*(z) = 0 on Th only if ( z ) is a critical point (z°) of Q°(z). But such a
critical point ( z° ) determines a solution (rj) of (8.1) satisfying the conditions
(8.10). If (z°) (0), the corresponding set
(Ml, •••,«„) = (Z?, ’ • ' , Z°)
cannot be null in (8.10) since (17) would be identically null. Thus if Q°(z°) — 0
on irh at a point (z°) ^ (0), the corresponding set ( u ) is not null. It follows from
(8.14) that Q{z) is negative definite on irh. The index of Q(z) is then at least h,
and since h ^ k the kth focal point of F must precede the fcth focal point of F°.
CHAPTER V
THE FUNCTIONAL ON A RIEMANNIAN SPACE
In classical problems in parametric form the domain of the variables is usually
taken as a region in a euclidean space. A more -general domain is a so-called
Riemannian space with a metric defined by a positive definite quadratic form
(0.1) ds2 = gi](x)dx,dxJ (i, j = 1, • • • , m).
We shall prefer a Riemannian space for two principal reasons. In the first place
a Riemannian space presents a suitable medium for a treatment of the numerous
invariants of the functional and for the presentation of the principal hypotheses.
In the second place the “ Jacobi least action integral’ ’ cannot be adequately
treated otherwise, at least in the large. For at the present time adequate
answers cannot be given to questions concerning the possibility of embedding
Riemannian manifolds in the large in euclidean spaces of high dimension.
A novel feature of this chapter is the invariantive formulation of the accessory
boundary problem. From the point of view of tensor analysis and Riemannian
geometry, entities may be regarded as geometrical if defined by invariants or the
vanishing of the components of tensors, because such entities are then inde¬
pendent of the coordinate systems employed. From this point of view char¬
acteristic roots as defined in this chapter are geometric entities. The classical
definitions of such roots do not afford roots of this character and considerable
care is required in the modification of the classical definitions. One calls an
entity restridedly topological if it can be defined by means which would be purely
topological except for analytical restrictions on the functions employed. It will
appear in Ch. VII that the number of negative characteristic roots is restrictedly
topological, at least in the non-degenerate case. This fact would be extremely
significant if one were to develop the present theory purely from the point of view
of abstract spaces, as presumably will be done shortly.
As a matter of detail we call attention to the considerable simplification in the
classical minimum theory due to the author’s elimination of Behaghel’s formula.
See Bliss [3]. We also give a proof of the existence of families of extremals cut
transversally by a manifold of any dimension. In general this chapter com¬
pletes the basic theory in the small.
A Riemannian space in the large
1. Riemannian spaces as ordinarily defined are local affairs. It is necessary
for us to add topological structure in the large. To that end we suppose that
we have given an ordinary ra-dimensional simplicial circuit K in an auxiliary
euclidean space on which a neighborhood of each point is well defined. See
Lefschetz [1], Veblen [1]. Our Riemannian space R will now be defined as
107
108
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
follows. Its points and their neighborhoods shall be the one-to-one images of
the respective points and their neighborhoods on K .• Moreover K shall be a
manifold in the sense that a neighborhood of each of its points shall be homeo-
morphic with a neighborhood of a point (x) in a euclidean ra-spaceof coordinates
(x) = (x1, • • • , xm). With at least one such representation of a neighborhood
of a point of R there shall be associated a positive definite ‘form such as (0.1),
defining a metric for the neighborhood. We suppose that the coefficients ga(x)
are of class C3. We term the coordinates ( x ) admissible. We also admit any
other set of local coordinates (z) obtainable from admissible coordinates ( x ) by a
transformation of the form
(1.1) zi — z'{x)
in which the functions z*(x) are of class C4 and possess a non-vanishing jacobian.
We also require that any two coordinate systems (x) and ( z ) which admissibly
represent a neighborhood of the same point P on R be related as in (1.1). In
transforming our problem to non-parametric form neighboring a given extremal
we shall find it necessary to admit transformations merely of class C3 and to term
the resulting coordinates specially admissible.
A set of points of R will be said to form a regular ? -manifold on R of class Cn if
the images of its points in any admissible coordinate system (x) are locally
representable in the form
= X*(UU • • * , Mr),
where the functions x'(u) are of class Cn in the parameters ( u ) and the functional
matrix of the functions x'(u) is of rank r. By a regular arc g of class Cn we shall
mean a closed segment of a regular 1-dimensional manifold of class Cn. By a
curve of class Dl we shall mean a finite continuous succession of regular arcs of
class Cl.
We shall now prove the following theorem.
Theorem 1.1. Let g be a simple regular arc of class C4 along which t is the arc
length. A neighborhood of g can then be admissibly represented as a whole by a
neighborhood of the xm axis in a euclidean space ( x ) in such a manner that g cor¬
responds to the xm axis with t = xm.
The theorem is true for a segment of g sufficiently near any point P of g. For
if (z) is an admissible coordinate system neighboring P, any segment of g suffi¬
ciently near P can be represented as stated in the theorem in the form z{ =
One at least of the derivatives ^(t) 9^ 0, say cpm 0. The transformation
(1.2)
Z ' = Xi + (p\xm)
Zm — <pm(xm)
(i = 1, • • • , m - 1),
is admissible and affords the desired local coordinate system (x). This leads
us to the following lemma.
[1]
A RIEMANNIAN SPACE IN THE LARGE
109
Lemma. Let there he given two overlapping segments gx and g2 of g, with g2
extending beyond gx and gx commencing prior to g2. If the theorem is true for g i
and g 2 separately , it is true for the arc gx + g2 into which gx and g2 combine .
For simplicity suppose that t = 0 is an inner point of both gx and g2. Suppose
that the coordinate system (x) represents g 2 as in the theorem, with t = xm
along g 2, and that the coordinate system ( 2 ) similarly represents gx with t = zm
along gx. Suppose that (x) = (2) = (0) when t = 0. Since both coordinate
systems are admissible neighboring the point t = 0 on g9 they are there related
by a transformation of the form
(1.3) zi = a)x3 + rj^x) (i,j = 1, • • • , m)
where a) is a constant, | a) | ^ 0, and rj'(x) is a function of class C4 with a null
differential at (x) = (0). Since t = zm — xm along g near the point t = 0, we see
that aZ ss 1. Without loss of generality we can suppose that a) equals the
Kronecker <5 ) since we could bring this about by replacing the coordinate system
(x) by the coordinate system
xl ~ a)x3.
Suppose then that (1.3) takes the form
(1.4) 2* = X{ + ^(x).
Let e be a small positive constant and /< (t) a function of class C4 in absolute value
less than 1 and such that
h(t) = 1, t ^ 0,
h(t) s 0, t ^ e.
The transformation (1.3) is valid neighboring t = 0 on gx and g2. If e is suf¬
ficiently small, the transformation
(1.5) 2* == x* + h{xm)n'{x ) {i =■ 1, • • • , m)
is defined neighboring the whole of g2. It is identical with the transformation
(1.4) for xm < 0, and reduces to the identity for xm > e. The coordinate system
(2) can now be regarded as representing the neighborhoods of gx and g2 combined.
Neighboring the points of gx for which t < 0, (2) shall represent R as before.
Neighboring points of g2 for which t ^ 0, (2) shall now be the representation
obtained from the given representation (x) by the transformation (1.5). Along
0i + 02 we see that zm — t. One also sees that the jacobian of the transformation
(1.5) is not null on g2 if e is sufficiently small.
To prove the theorem we cover the whole of 0 by a finite ordered set of local
coordinate systems each of the required nature neighboring the portion of g
covered, and excepting the first, each overlapping its predecessor neighboring
no
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
some point of g. The theorem follows upon making a finite number of applica¬
tions of the preceding lemma.
We shall prove the following theorem.
Theorem 1.2. There exists a non-singular transformation of class Cz of the
coordinate system of Theorem 1.1 into coordinates (x) in which g again corresponds
to the xm axisf while xm equals the arc length t along g and ga{x) — b) along g.
Let the coordinates of Theorem 1.1 be denoted by (z). Along g then zm — t.
Let aij(zm ) be the value of <7,,(z) at the point t — zm on g. We make the trans¬
formation
x{ — z* (i — 1, • • • , m — 1),
(1.61
xm = amj{zm)zJ (j = 1, • • • , m)y
generalizing the reduction of quadratic forms due to Lagrange. See Bocher [1],
p. 131. We note that amm = 1, since t = zm along g. We then see that along g
the differentials of (x) and (z) are transformed in the same manner as ( x ) and
(z)7 and that accordingly (1.6) carries the basic form ds 2 into one in which along
gf dxm appears only in the form ( dxm )2. As in the Lagrange reduction we turn to
the residual form in the variables dxl, • ■ • , dxm~l. By making transformations
similar to (1.6), this residual form can be reduced along g to a form in which the
squares of the differentials alone appear, multiplied by positive functions of zm.
A further obvious transformation will make these coefficients unity along g.
Thus ds2 will take the required form. Moreover after each transformation t — xm
as desired, and the theorem is proved.
The coordinate system of Theorem 1.2 will he termed normal. This system is
specially admissible.
Up to this point we have supposed that the arc g is without multiple points.
In case g has multiple points we divide it into a finite number of consecutive
segments so small that each arc of g which is composed of three successive seg¬
ments is without multiple points. According to the preceding theorems, each
of these segments is interior to a normal coordinate system. We restrict these
coordinate systems to neighborhoods N of the segments so small that no system
has any points in common with the second following or second preceding system.
We now define a new Riemannian manifold covering R on which each of the
above neighborhoods N is to be regarded as distinct from each of the other neigh¬
borhoods save the ones immediately following and preceding N. On the
resulting manifold R the arc g is without multiple points and possesses a neigh¬
borhood coverable by a single normal coordinate system.
The definition of a Riemannian manifold here used was introduced in brief in
1929 in Morse [4], p. 166. Veblen and Whitehead (see Veblen [2, 3]) have
presented a general axiomatic basis for differential geometry. The manifolds
which we employ come under those of Veblen and Whitehead, although the
definition adopted in this chapter is perhaps simpler for our purposes.
[2]
BASIC TENSORS
111
Basic tensors
2. With each admissible coordinate system (x) we suppose that we have given
a function
F(x, r) = F(x\ ■ ■ ■ , xm, r\ ■ ■ ■ , rm) (r) ^ (0)
serving to define an integral
(2.0) / F(x, x)dt
in that system, where x{ represents the derivative of xi with respect to the
parameter t. When t he variables (x) are subjected to the transformation
(2.1) s’* = *'(*),
we understand that the variables (r) are subjected to the transformation
(2.2) ch = r* (h, i = 1, • • • , m).
That is, we suppose t hat (r) is transformed as a contravariant tensor or vector.
The function F(x, r) is then to be replaced by a function Q(z, a) such that
(2.3) Q(z, a) = F(x, r), (a) ^ (0)
subject to (2.1) and (2.2). Our integral then takes the form
/ Q(z, e)dt .
Fdr at least one coordinate system neighboring each point of R (and con¬
sequently for all such coordinate systems) we assume that the corresponding
integrand is positive and of class C3 for (r) ^ (0). We also assume that F(xy r)
is positive homogeneous of order 1 in the variables (r). That is we assume that
(2.4) F(: r, hr) - kF(x, r), (r) ^ (0),
for all positive numbers k . Upon differentiating the identity (2.4) with respect
to k and rj we find that
(2.5) r'F rirj = 0 (i, j = 1, • • • , m),
so that
I Fm | S 0,
an important peculiarity of the parametric form.
We shall now consider certain tensors and invariants which enter into the
theory. See Eisenhart [1].
Upon differentiating (2.3) with respect to r\ with ch subject to (2.2), we find
that
112 THE FUNCTIONAL ON A RIEMANNIAN SPACE [ V ]
Thus Fri is a covariant vector. It follows that
(2.7) Fridxi
is an invariant. The expression (2.7) enters into the transversality conditions
and into the Hilbert integral. If (r) and (a) are contravariant tensors, the
Weierstrass .E'-function
(2.8) E(x , r, a) = F(x, a) - a{Fri(xy r)
is another invariant. The expression
is an invariant provided (X) is a contravariant vector, since Frij is seen to be a
covariant tensor of the second order. One also verifies the fact that
dzk [~ d
dXi _dt
Qak
We consider the bordered determinant
(2.9)
We see that
F
1 r'r)
*>i
Ui
0
= B.
(2.10)
Frirj Ui
V) 0
AxlUiVj
where A'3 is the cofactor of FrirJ in the determinant of these elements. But if
the first m rows of B are multiplied respectively by rl, • • • , rm, added, and then
substituted for the fcth row, the elements in the resulting fcth row will all be zero
by virtue of (2.5), provided
r'Ui = 0.
Regarding B as a polynomial in Ui and Vj we see that r*Ui must be a factor of By
at least if rk ^ 0. But we are assuming that (r) ^ (0), so that at least one of
the variables tk ^ 0. By operating upon the columns of B in a similar manner
we see that rh, is also a factor of B. Thus we have the relation
(2.11) A^UiVj ss Fi(z, r) [r^J [r3Vj]
where F i is a factor of proportionality. If we let ( u ) = ( v ) = (r), we see that
F i(x, r)
A'W'r1
(r*r‘)2
so that F i is continuous in (x) and (r) for (r) 5^ (0).
[3]
NECESSARY CONDITIONS
113
Upon equating the coefficients of u&y in the two members of (2.11), we find
that
(2.12) Aij = Fir**,
relations which have been used by Weierstrass to define F i when m = 2, and
by others when m > 2. See Hadamard [1], p. 95, and Bliss [3].
If ( u ) and ( v ) are transformed as covariant vectors into vectors ( u' ) and ( v'),
it follows from the theory of adjoint quadratic forms (see B6cher [1], p. 160)
that
F
1 r'rJ
Ui Q<ri<ri
= C2
Vi
0 v\
0
where
dz{
c = — .
dxJ
From (2.11) we then see that
Fi(xy r) = c2Qr{zy a)
where Qi is formed from Q in the system (zy a) as was from F in the system
(x, r). See Bolza [1], p. 346.
The necessary conditions of Euler, Weierstrass, and Legendre
3. Suppose that we have given a simple, regular, sensed curve g of class C 1
joining two points A1 and A2 on R. We admit sensed curves of class Dl joining
the end points of g in the order A1 and A 2, and denote the value of our integral
along such curves by J. We state the following theorem.
Theorem 3.1. If g affords a weak relative minimum to Jy it is necessary that
(3.1) Fri - Fzi - 0 (» = 1, •••,«)
along g in each coordinate system (x) in which g enters.
To prove this theorem we turn to a particular coordinate system (x) in which g
enters, and consider the problem of minimizing J as an integral in non-parametric
form in the space of coordinates ( ty x1, • • * , xm). If
2^ = y*(t) (tl £ t ^ t2),
is a representation of class Cl of an arc of g in the system (x), these same equations
define an arc g in the space (f, xl, • • • , xm). If g is a minimizing arc in the space
(x), g will afford at least a weak minimum in the corresponding non-parametric
fixed end point problem in the space ( ty x). Conditions (3.1) then follow from
the non-parametric theory.
114
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
A regular arc which is of class C2 and satisfies (3.1) will be called an extremal .
The condition of Weierstrass is as follows.
Theorem 3.2. If g affords a strong , relative minimum to J , it is necessary that
E (Xj x, a) ^ 0
for ( x , x) on g and for any non-null vector (<r).
To prove this condition we again operate in a single coordinate system (x).
Suppose the parameter t on g so chosen that t = 1 specifies a prescribed point
P A1 on g9 and the points 0 ^ t g 1 all lie on g in the coordinate system (.r).
Suppose that g is cut at the point t — 1 by a regular curve x * — xx(a) of class
C1 for a ^ 1 and near 1, and such that
7*(1) = *f0)
where x 1 == y'(t) is our representation of g. We evaluate J along the curve,
a = const., of the family,
x% a) = y\t • a) + (x^a) - y{(a))t (0 S t £ 1),
passing from the point t = 0 to the point t — 1. To this we add the value of J
along the curve xx = x\a) passing from a point a < 1 to the point a = 1. We
see that
X^t, 1) see y *(«),
X»(0, a) = 7*'(0),
£*(1, a) = £*(«),
and with the aid of these identities we find for the function J(a) that
j'( 1) = xKl) Fri[ 7(1), 7(1)] - F[7(l), i(l)].
But for a minimizing arc it is necessary that J'(l) g 0. We set x‘(l) = a' and
observe that (a) can be taken as an arbitrary non-null vector. The condition
J'( 1) ^ 0 reduces to the condition of the theorem in case P ^ A1.
The continuity of E insures the truth of the theorem in the case P — A1 as
well.
As a corollary of the Weierstrass condition we have the following analogue of
the Legendre condition.
Corollary. For a weak minimum it is necessary that
Fr%rj(Xj z)X‘V g; 0
for ( x , x) on g and for any vector (X).
To prove this corollary we consider the function
<p(e) = E(x(t)j x(t)y x(t) + eX)
[4]
EXTREMALS
115
where x * = x'(t) defines g. Observe that <p(e) has a minimum zero, when e = 0,
by virtue of the Weierstrass condition. Hence v?"(0) ^ 0. But a simple com¬
putation shows that <p"(0) is the left member of (3.2), and condition (3.2) is
established.
Extremals
4. We shall continue by obtaining a general representation of extremals. To
that end we first verify the fact that the relations
(4.1)
(r* = £*)
hold identically along any regular curve — x'it) of class C2. These identities
are a consequence of (2.5) and the identities
r'F rixj = Fxi (i,j = \, ■ ■ ■ ,m)
derived from (2.4) by differentiation with respect to k and xJ.
We set
<p(x, r) = gaix^r’
and consider the differential equations
(4.2)"
<p(*y *) = 1,
of which (4.2)" requires t to be the arc length. Let \{t) be an unknown function
of t and consider the system
(4.3)' ~ (Fri + Vri) “ (Fxi + V*t) = 0,
(4.3)"
<p(x, x) = 1.
Upon differentiating (4.2)" with respect to t and making use of the homo¬
geneity of <p(x, r) in the variables (r), one finds that
(4.4)
(r* _ j.iy
Upon multiplying the zth condition in (4.3)' by x\ summing, and making use of
(4.1) and (4.4), we find that
zVVi
dX
dt
4
dt
= 0.
Hence for any solution x'(t), X(<), of (4.3) for which X = 0 initially, we must have
X 3. 0. The functions x'(t) will then define a solution of (4.2).
116
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
Suppose that we have a solution g of (4.2) in the form
x< = y‘(t)
(i = 1, • • • , m)
where the functions y *(<) are of class Cl on an interval (tl, t2). It will be con¬
venient to set
f '!(<) = Fr.(y{t), i(t))
and to term 7 *(/), y'(t)9 v°i(t) sets x\ r\ v9 on g. We assume that F 1 ^ 0 for
( x , r) on g . From (2.10) and (2.11) it then follows that
(4.5)
F
1 r*rj
S Pri
<Pri
0
7* 0
on g.
To solve the equations (4.3) we set
Vi = Fri(x , r) + Xv?r*(x, r),
1 = <p(x, r).
By virtue of (4.5), equations (4.6) have unique solutions
r* = ^(ar, *>)>
X = X(x, i>),
(4*6)
(4*7)
of class C2 in (2) and (v) for (2, r, v) near sets on g, and X near zero. The system
(4.3) can then be given the form
(4.8)
where
dxx v N
w = r'(x’ v)’
= qt{x, v), X = X(ar, v),
*7« = Fxi[x, r(x, v)] + X(ar, v)<pxi\x, r(x, »)].
Equations (4.8) have solutions of the form
(4.9)
= h'(t, to, x0, Vo),
Vi = ki{t, t0, x0, v0),
which take on the values (x0, v0) when t = U, and for which the functions h •
and k, are of class C2 in their arguments for t and to on the interval (tl, t2) and
(t0, x0, vo) sufficiently near sets (t, x, v) on g.
We do not wish the general solution of (4.3) but only those solutions which
are solutions of (4.2), and are solutions for which X = 0 initially. We obtain
these solutions from (4.9) by setting
(4.10)
ViO = Fri(x o, r0)
[5]
CONJUGATE POINTS
117
in (4.9), since (4.10) taken with (4.6) is easily seen to imply the condition that
X(x0, *>o) = 0. Our general solution of (4.2) thus takes the form
(4.11) x* = x^t, t0, xq, r0)
where the functions on the right are of class C2 in their arguments for t as before
and (*o, xo, r0) sufficiently near sets x , ±) on g . Moreover we have
x'q = x^to, to, x0, r0), <p(x o, ro) = 1,
i o ~ to, Xo, r0).
Reference to the first of equations (4.8) discloses the additional fact that the
functions ki and hence the functions x\ are of class C2 in their arguments.
Suppose that (z) is a coordinate system overlapping the system (x). In the
system ( z ) suppose there is given a family of extremals neighboring g , with t as
the arc length, depending on certain parameters (a). To continue this family
in the system (x) we understand that a point t = is selected on g which lies in
both the systems (x) and (z). By means of the transformation between the
two coordinate systems the values of the variables (x, x) at the point t = ti on
the extremal determined by (a) can be expressed as functions xj (a), rj(a). We
then regard the family
x*' = x*(£, tly xx (a), r^a)) = ^(t, a)
as a continuation in the system (x) of the family of extremals given in (z). It is
clear that the functions ip'(t, a) will be independent of the particular point t = ti
on g used to define them.
Conjugate points
5. We shall define the conjugate points of a point t = t\ on an extremal g.
Let (p) be the unit contravariant vector which gives the direction of g at t = t\.
Suppose the point t = t\ on g is interior to a coordinate system (x). In the
system (x) let the components r* of the unit vectors neighboring (p) be regularly
represented as functions r'(u) of class C2 of n = m — 1 parameters (u). Suppose
that (p) corresponds to (u) = (0). In the system (x) the extremals issuing from
the point
x'l = T‘(<i)
on g with directions neighboring (p) can be represented in terms of the functions
in (4.11) in the form
(5.1) x< = x‘(t, tlt y(h), r(w)> = ^(t, u ).
The jacobian
M(t, h)
Z)(yl, • • • , y”)
Ulj / Un)
(5.2)
(u) * (0),
THE FUNCTIONAL ON A RIEMANNIAN SPACE
118
[V]
vanishes at t = tx. We can factor t — tx out of each of its last n columns, and
for t neighboring t = tx represent this jacobian in the form
(5.3) M(t , /i) = (t — *i)n N(t, tx) (n = m - 1)
where <i) is continuous in / and tx and
(5.4) N{t, h) = | p\ r;.(0) I (t == 1, • • • , == 1, • • • , n).
The last n columns of this determinant give n independent vectors, since the
vectors r*(u) are regularly represented. Moreover these columns represent
vectors orthogonal to the vector (p) as one sees upon differentiating the identity
s 1 (i,j = ], * • * , w)
with respect to w* and setting (u) = (0). Thus JV(<i, L) 5* 0. Hence near
t = the determinaht Af(f, U) vanishes at most at t — t\.
If the family (5.1) is “continued” into an overlapping coordinate system ( z)f
one obtains a new local representation of these extremals and a new jacobian
D(*1, • - , *m) ^
I^(^j U]f f ^n)
(u) = (0),
which we call a continuation of the original jacobian. On any common segment
of g these jacobians will vanish simultaneously and to the same orders in t.
By the conjugate points of the point t = tx on g we mean the points t t\ on g at
which the jacobian M(t, tx) or its successive continuations vanish .
Let g be an extremal on which the arc length t increases from tl to t2 inclusive.
To show that the conjugate points of t = tl are isolated we need to represent our
functional in non-parametric form. To that end we refer the neighborhood of
g to normal coordinates (x) as in §1, with g corresponding to the xm axis and
t = xm along g. We set
and
(*S • • • , am) = (yh ■ ■ • , yn, x) (n = m - 1)
(5.5) f(x, y yn, pu •••,?») = F(y u ■ ■ ■ , yn, x, ph ■ ■ ■ , pn, 1)
where F(z, x) is the integrand in the normal system ( x ). For any admissible
curves neighboring g for which xm > 0 our functional J takes the form
J = J' f(x, y, y')dx (I1 ^ i ^ P).
We assume that the Legendre S-condition holds along g, that is that
(5.6) Fr<rl(x} ±)\<V >0 (i, j = 1, ■■■ , m)
for (x, x) on g and for any unit vector (X) not ±(x). For the present system of
coordinates we see that
Frmrl(x, x) = 0
(j = 1, ■■■, m)
[6]
THE HILBERT INTEGRAL
119
along g, as follows from (2.5). Moreover we see from (5.5) that
Priri = fPiPj (i, 3 = 1, • • • , m - 1)
so that (5.6) takes the form
fpipfz’ > 0 (h 3 = 1. ■■■ ,m - 1)
for sets (z) ^ (0). Thus the Legendre iS-condition holds along g in the non-
parametric problem.
As in (5.1) let the extremals through the point t — tl on g be represented in the
form
xi = <p% u ) (t1 g t g t2)
with (u) = (0) corresponding to g. Since
MU 0) = 1 * 0,
we can take xm as a parameter instead of t, and so represent these extremals in
the form
Vi = ]/i(x9 u).
One sees that for t — x,
T)(<p\ • • • , <fm) _ />(?/!, • • , !/v)
I)(t, Ui, • • • , Unj • • ■ , ttj ?
(“) = (0).
This is an identity in t = x. Nowt the first determinant in (5.7) vanishes near
t = tl , at most at t = J1. The right hand determinant certainly does not then
vanish identically near £ = t1. But its columns represent a base for a conjugate
family of solutions of the Jacobi equations in non-parametric form. We draw
the following conclusions from the theory of conjugate syslems. Cf. Oh. Ill, §3.
If the Legendre S-condition holds along g} the conjugate points of a given point are
isolated , and the jacohian M(t , t\) and its continuations , defining these conjugate
points by their zeros} vanish at most to the order m — 1 in i .
The Hilbert integral
6. Let A be an n-parameter family of extremals represented in terms of the
arc length t and m — 1 parameters (0). We suppose that t ranges over an
interval tl g t g t2 and that (0) is a point in an open simply-connected region
in a euclidean n-space. Locally we suppose that the points on A are repre¬
sentable in the form
(6.1) x{ = *•'(*, P),
where the functions x'(t, 0) are of class C2 in their arguments for (ty 0) near some
particular set ( t° , 0°), and where
D(x\ • • • , x”) Q
D(t, Hi, • • • , Hn)
(n — m — 1).
120
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
In the large we assume that the totality of points on A forms a one-to-one
continuous image on R of the complete product domain ( t , fi). We say then
that the extremals A form afield Si.
Locally the parameters ( t , 0) corresponding to a point (x) of Si will be functions
t(x)} fiK(x) (h = 1, • • • , n),
of class C2. We set
r'(x) = *;(<(*), &(*))■
Locally the Hilbert integral then has the form
jFri(x, r(xj)dx\
Essentially as in Ch. I, §6, we could prove the following:
A necessary and sufficient condition that the Hilbert integral be independent of
the path on the field Si is that on this field
wXF-w)-UK‘w) <*■*
In particular the family of extremals (5.1) passing through the point t — tl on g and
neighboring g} forms such a field neighboring any segment of g which is simple and
closed in the point set sensey on which t t\ and on which there is no conjugate point
of t = tl.
A field Si on which the Hilbert integral is independent of the path will be
called a Mayer field .
Sufficiency theorems
7. We begin by enumerating certain conditions which appear in subsequent
theorems. In all of these conditions we suppose that we have an extremal g
on which the arc length t increases from tl to inclusive, and which is locally
representable in the form x * = No generality is lost by assuming that g
is without multiple points, for in the case of multiple points we have seen in §1
that we could cover the neighborhood of g by a new Riemannian space N in
which g would be replaced by a covering extremal without multiple points.
By the Jacobi S-condition is meant the condition that there be no conjugate
point of the initial point of g on g.
By the Legendre S-condition on g is meant the condition that
(7.1) Friri(xy *)X‘V > 0 (i,j « 1, ... ,m)
for (x, x) on g and (A) any unit vector not db (x).
By the Weierstrass S-condition on g is meant the condition that
(7.2) E(z, x, <r) > 0
for (x, x) on g and (<r) any unit vector not (x).
[7]
SUFFICIENCY THEOREMS
121
The problem will be said to be positive regular on a domain S of R if for each
local representation of points (x) of $
(7.3) FrirJ(x, r)X'V > 0,
for (x) on S and for arbitrary unit vectors (r) and (X) of which (r) is not zk (X).
We begin with the following lemma.
Lemma 7.1. If Fi 5* 0 along g and the Weierstrass S-condition holds along g,
then in each local system
(7.4) E{x, r, a) > 0
for all sets (x, r, <r) in which (r) and (a) are unequal unit contravariant vectors and
the set (x, r) is in a sufficiently small neighborhood of the sets (x, x) on g .
Without loss of generality we can suppose that g is covered by a single co¬
ordinate system (x). Let the condition (7.1) with the equality added be denoted
by (7.1) . We note first that (7.1)' must hold as a consequence of the Weier¬
strass ^-condition. But from the condition that Fx 9^ 0 along g it follows that
j FrirJ | has the rank m — 1, as one verifies from (2.12). From this fact and
(7.1)' it follows that (7.1) holds.
Now the roots p of the characteristic equation
(7-5) I fU,-p5' I = 0
with (x, r) on g, are all positive by virtue of (7.1), save one which is null, cor¬
responding to the fact that | Frir}- | is always null. But from the continuity of
these roots we see that the same state persists for (x, r) sufficiently near sets on p.
Consequently (7.3) must hold for sets (x, r) sufficiently near sets on g and (X) any
direction different from =t (r). We turn to the definition of the Weierstrass
F-function and use Taylor's formula to represent E as a function of (a), with
(r) as the point of expansion. We see that (7.4) must hold when (r) and (a) are
unequal unit vectors and (x, r, a) is on a sufficiently small (open) neighborhood
N of sets (x, x, x) on g.
But sets (x, x, <r) not on N, for which (a) is a unit vector and (x, x) on g form a
closed ensemble for which E is bounded away from zero by virtue of (7.2).
Hence for sets (x, r, a) not on N for which (r) and (cr) are unit vectors, and (x, r)
within a sufficiently small neighborhood N\ of sets (x, x) on g) E will still be
positive.
The lemma holds as stated for (x, r) on N\.
We come to the following theorem.
Theorem 7.1. In order that the extremal g afford a proper , strong minimum to J
relative to neighboring curves of class D1 which join g’s end points , it is sufficient that
the Weierstrass and Jacobi S-conditions hold and F 1 5^ 0 along g.
By virtue of the preceding lemma the condition (7.4) holds in an easily applied
form. To obtain a Mayer field including g we make use of the identity of con-
122
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
jugate points in the parametric theory with those in the non-parametric theory,
and infer that the extremals issuing from a point on g prior to g’ s initial point
A1, but sufficiently near A1, form a Mayer field covering g. From here on the
proof is essentially the same as the proof of Theorems 7.1 and 7.2 of Ch. I.
We shall establish the following corollary of the theorem.
Corollary. In order that g afford a proper , strong minimum to J it is sufficient
that the Jacobi S-condition hold along g , and that the problem be positive regular
along g.
We shall show that the hypotheses of the theorem are fulfilled under the
conditions of the corollary. In particular under the condition of positive
regularity | Frirj | must be of rank m — 1 for (x, r) on g, and hence Fx ^ 0 on g .
To deduce the Weierstrass ^-condition from the regularity condition we turn
to the function E(x, r, <r) and let (r) and (a) be any two unit contravariant vectors
such that (r) is not -f-(a). A use of Taylor’s formula as described in the proof
of Lemma 7.1 now shows that E(x , r, a) >0 for (r) not — (<r). This is a conse¬
quence of the regularity condition. For (r) = —(a), Taylor's formula is not
applicable since F is not defined for (r) = (0) and such a point might enter in the
application. But this difficulty is easily met. One verifies the fact that the
identity
(7.6) E(x, r, - r) = E(x, p, - r) + E(x, p, r)
is valid for any two unit vectors (p) and (r). If (p) is now chosen different from
zb (r), the right member of (7.6) is positive by virtue of (7.3). Hence
E(x, r, - r) > 0
for (x) on g. Thus the Weierstrass ^-condition on g is implied by the condition
of positive regularity along g.
The corollary follows from the theorem.
Note. In order to meet the difficulty that Taylor’s formula could not be
applied for (r) = — (a), a formula known as BehaghePs formula has been intro¬
duced. In the light of the above treatment this formula is no longer necessary.
The Jacobi equations in tensor form
8. Let g be an extremal locally representable in a coordinate system ( x ) in
the form
(8.1) x{ = y*(t)
where t is the arc length along g. Corresponding to g we set
2co(t7, 1)) = Frirfii'?i’ + 2 FriXJiiirji + FrixjV V (i, j = 1, • • • , m)
where the arguments (x, r) in the partial derivatives of F are taken on g. Let
x' = x*(t, e) be a family of curves joining the points t] and f on g in the system
[8]
THE JACOBI EQUATIONS IN TENSOR FORM
123
( x ) and reducing to g for e = 0. Suppose that the functions xl(t, e) are of class
C2 for t on (tl, t 2) and e near 0. The second variation takes the form
(8.2) J"(0) = j‘ 2u(v, i))dt [r = x\(t, 0)].
If we change from coordinates (x) to coordinates (z), we naturally understand
that tj* shall be replaced by the variation vi(t) along g of zi with respect to e .
The variation (77) is then transformed as a contravariant vector. Thus
r
y'(t) = -T- ’Jo (0 (i, j = 1, ■■■, m)
where the partial derivatives of (x) are evaluated at the point t on g. In terms
of the integrand Q(z, z) replacing F(x, x) in the system (z), we set
2a)°(rfo) 7)0) ^ 0^0 T“ j "T Qz'zjVoVb
evaluating the partial derivatives of Q along g as before. If r^i) and rjo(t) are
components of class C2 of the same contravariant vector given respectively in
the systems ( x ) and (z), I say that
(8.3)
d
dt ^ ~
] ( d 0 __ 0 \
dx'\dt “V./
where the partial derivatives of z1 are evaluated at the point t on g.
To establish (8.3) consider the family of curves
x{ = x'(t , e) = 7l(0 + erjx(t),
and let z'(t, e) represent the same family of curves in the system (z). As we have
seen in §2, we have
(fu) £ [a «--«■*]
where we understand that
xi = x*(t, e), r{ = x\(ly e), e)y <j{ = z)(t, e).
Equations (8.3) follow from (8.4) upon differentiating (8.4) with respect to e
and setting e = 0.
Thus the operator'
Li(v)
00 ijt
is a covariant vector provided g is an extremal.
The Jacobi equations are not all independent . In fact they satisfy the relation
(8.5) 55 0,
an identity in t for all sets (77) of class C2 in i.
124
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
To prove (8.5) we make use of the previously established identity
In particular we set
x{ = yi(t) + ey{(t)f
** = 7*(0 + «44(0»
whereupon (8.6) becomes an identity in t and e. Upon differentiating this
identity with respect to e and setting e = 0, (8.5) results as stated.
We come to the question of the solutions of the equations L<(i/) = 0. The
determinant of the coefficients of the variables i j* in L»(r?) is I Frirj |) and is
therefore null. To meet the difficulty which thereby arises we replace the equa¬
tions Li( tj) = 0 by the system
(8.7) ' Liir,) = 0,
(8.7) " ^ (ff.yyV) = 0 (i,j = 1, • • • , m).
The parenthesis in (8.7)" is an invariant which we denote by yT. It is the
algebraic value of the projection of the vector ( y ) on the tangent to g at the point t.
From (8.7)" we see that
y = at + by
where a and b are constants.
To solve the system (8.7) we introduce the auxiliary system
(8.8)
Li(v) + = 0,
d2 , .. A
di* {9iiy v) = °-
The determinant of the coefficients of the variables ij' and n in (8.8) is seen to be
Qai*
OifY’
0
— F\(y, i) ** 0.
Use has thereby been made of (2.11). We can solve the system (8.8) for the
variables ij* and n in terms of the remaining variables (t, ij, i)) in (8.8). But upon
using (8.5) we see that n = 0 in solutions of (8.8), so that (8.8) may be regarded
as identical with (8.7). Accordingly (8.7) can be put in the form
V* - M^t, rj, 1)
where M(t, y, if) is linear and homogeneous in the variables y* and ii*.
[8]
THE JACOBI EQUATIONS IN TENSOR FORM
125
In the conditions (8.7), LJrj) is a covariant and the parenthesis in (8.7)" is an
invariant. By the Jacobi equations in tensor form along g we understand a set of
conditions of the form (8.7) for each local coordinate system (x) into which g
enters. By a solution of these equations we mean a contravariant vector defined
along g with a representation ^(t) of class C2 in each coordinate system in which
g enters, satisfying the corresponding system (8.7). The identity of two solu¬
tions is conditioned then merely by the identity of the contravariant vectors
which define these solutions. A set of solutions are dependent if their representa¬
tives v ‘(0 in each coordinate system are dependent. It is clear that dependence
in one system necessitates dependence in all.
With this understood we state the following theorem.
Theorem 8.1. A point t = t" on g is conjugate to a point t = V on gif and only
if there is a solution of the Jacobi equations in tensor form which is not identically
null and which vanishes at V and t". Moreover the number of independent solutions
vanishing at tr and tn equals the corresponding numbei in a non-par ametric repre¬
sentation of the problem in normal coordinates .
To prove the theorem we refer the neighborhood of g to the normal coordinates
of §1. The extremal g is thereby represented by the xm axis and ga = d] along g.
Along g we have
VT = Qifi'V3 = vm
so that the condition (8.7)" here implies that
rjm — at + b.
Accordingly a solution (77) of (8.7) which vanishes twice must here be such that
7?m ss 0. Moreover reference to (8.5) shows that the condition Lm(rj) = 0 is
always satisfied by sets (77) of class C2 so that it may be discarded. Accordingly
for solutions of (8.7) which vanish twice (8.7) reduces to the conditions
(8.9) LJrj) — 0, ?7m = 0 (i = 1, • • • , m — 1).
Suppose the problem is now put into non-parametric form as in §5 with
f(x, y , y') as the integrand, and
(8.10) A = 0 (* = 1, ■ • • , m - 1),
the corresponding Jacobi equations set up for the x axis as an extremal with
dependent variables
(8.11) rj\ • • • , *?n (n * m - 1).
Using (5.5), we verify the fact that if t = xy the conditions (8.9), in so far as they
bear on the variables (8.11), are identical with the conditions (8.10). The
theorem follows directly.
126
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
The general end conditions
9. We suppose that we have given an extremal g on which the arc length t
increases from tl to t2 inclusive. Points near the initial and final end points of g
will be denoted respectively by
(a:11, • • • , xml) (a;12, • • • , xm2).
A curve of class Dl neighboring g will be termed admissible if its end points are
given by functions
(9.1) xia = x'*(a1, • • * , ar) (0 g r g 2m; s = 1, 2)
for values of the parameters («) near (0). For r — 0 we understand that the
functions on the right symbolize the end points of g. For r > 0 and for (a)
near (0) we suppose the functions in (9.1) are of class C2 and that they give the
end points of g when (a) = (0).
For r > 0 let 0(a) be any function of (a) of class C2. For r = 0, 0(a) shall
represent the symbol 0.
Our general functional now has the form
J — 6(a) + jF(x , x)dt
where the integral is to be evaluated along admissible curves with end points
determined by the set (a).
Our transversality condition here takes the form
(9.2) rf* + («) = (0),
where (x, r) is to be taken on g at the respective ends of g. The differentials
dd and dxi8 are to be expressed in terms of the differentials dah> and (9.2) regarded
as an identity in these differentials. We shall now prove the following:
A necessary condition that g afford a weak minimum to J relative to neighboring
admissible curves of class C 1 is that it satisfy the transversality condition (9.2).
The end conditions impose no restrictions on the end values of the parameter t.
In particular we will certainly still have a minimum if we restrict ourselves
to admissible curves for which the end values of t are tl and t2. If we regard the
problem as one in the space of the variables ( t , xlf • ♦ • , xm) in non-parametric
form, the above transversality condition follows from the corresponding condi¬
tion in non-parametric form.
The second variation
10. We have already obtained a formula for the second variation in the case of
fixed end points. We consider the case r > 0. We again suppose the neigh¬
borhood of g covered by a single coordinate system (a;).
Suppose that we have given a set of functions ah(e) of class C 2 for e near 0,
and a 1-parameter family of curves
x{ — x{(t> e)
(tl £ t £ tl)
127
til] THE ACCESSORY PROBLEM IN TENSOR FORM
such that x{(/, e) is of class C 2 for t on its interval and e near 0, and such that
(10.1) x'(taj e) 55 xia(a(e )),
(10.2) x*(t, 0) s t *(*),
where 7^) defines g. For each value of e near 0 we evaluate J = ,7(e) along the
-corresponding Curve (10.1), taking 0 as 0(a(e)). We readily find that
J’ + f‘t Mv, m,
where = x‘(/, 0), (a) — (0), and (x, r) in Fr» is to be taken on g at the respec¬
tive ends of g.
We indicate differentiation of the functions xl*(«) and 6(a) with respect to
cth or ak and evaluation at (a) = (0), by subscripts h or k respectively. If we
set c**'(0) — uh, the second variation takes the form
(10.3) J"( 0) = + [* Mv, t)dt (h, k = 1, • • • , r)
where
(10.4) fe=M0) + [fr4(0)]|.
Moreover if we differentiate (10.1) with respect to e and set e = 0, we find that
(10.5) Tji8 — x™(0)ah = 0 (i = 1, • • • , m; h = 1, • • * , r; s = 1, 2),
where the superscript 5 on ( rj ) indicates evaluation at t = t‘.
As in the non-parametric theory in Ch. II, §1, so here we are led to consider
the functional (10.3) subject to (10.5). We term the conditions (10.5) the
secondary end conditions. If a curve ? f(t) of class C1 and set ( u ) satisfy (10.5)
and afford a minimum to the second variation among curves of class Dl and sets
(u) which satisfy (10.5), it is necessary that ? f(t) satisfy the Jacobi conditions
Li(i 7) = 0 and a counterpart of the transversality conditions of (9.2). If we set
U = ’))»
these transversality conditions take the form
(10.6) f M2 - f + PhkUk = 0 (i = 1, • • • , m; h, k = 1, • • • , r)
where stands for the value of f »■ when t = t \
We term (10.6) the secondary transversality conditions .
The accessory problem in tensor form
11. In order to define the accessory problem in tensor form we introduce
certain new tensors. To that end we let x{ = y'(t) represent the extremal g as
previously, with t the arc length. Let 77' be a contravariant vector at the point
128 THE FUNCTIONAL ON A RIEMANNIAN SPACE [ V ]
t on g. The covariant components 77* of this vector and of the vector 7* have the
respective forms
Q<n\ gai’-
Let us resolve (77) into components tangent and orthogonal to g respectively.
The algebraic value of the component of (77) tangent to g at the point t is the
invariant
(11.0) tjr = (p, q = 1, • • • , rn).
The covariant vector projection of (77) on the tangent to g at the point t is then
vTi = vTQni’-
Let (77*) represent the covariant component of (77) orthogonal to g . We have
_ r
v% — V* Vi-
Combining the preceding results we obtain the formula
(1L1) v < = gov’ - (gaiO to«rV)
giving 77* as a linear function of the variables 77*.
Our accessory problem is now formally defined by the conditions
(11.2a)
vr - 0,
(11.2b)
Li(v) + = 0
(»”!,••■
(11.2c)
-a
1
H
£
»r
II
o
(s =
1,2),
(11. 2d)
42r< - x\Yi + /w = o
(h, Jc = 1, • •
• , r),
where 77 r and 77* are given by (11.0) and (11.1) respectively.
The conditions of this problem are well defined and self-consistent if they are
associated with a single coordinate system (x) covering the whole neighborhood
of g. They are also well-defined if different coordinate systems are used. For
the left member of (11.2a) is an invariant, the left members of (11.2b) define a
covariant vector, those of (11.2c) a contravariant vector, and finally the left
members of (11. 2d) are invariants, subject to (11.2c), as we shall see.
To that end we write (11. 2d) more fully in the form
(11.3) (afrS + F\&u*) - (afri + FUx&u') + 0Aiu* = 0.
The term 6hkUk is clearly an invariant. We shall prove that the second paren¬
thesis is an invariant subject to (11.2c). That the first parenthesis is also an
invariant subject to (11.2c) will follow similarly.
The statement that the parentheses in (11.3) are invariants is not yet well-
defined in that we have not yet stated how f * is to be transformed. To come to
this point let y* be the components of a contravariant vector associated with the
[11]
THE ACCESSORY PROBLEM IN TENSOR FORM
129
point / on g in a system (x).
system (z). We have
(11.4)
Let 77 J be the components of the same vector in a
where the partial derivatives are evaluated at the point t on g. When 7 7** and
if* are formally given at a point t on g , we understand that 1)0 is then defined in the
system (z) by the equations
where the coefficients of 1 y and rj3 are taken along g. This is consistent with the
behavior of actual variations. With this understood the variables and f*
in the systems (x) and ( z ) respectively are defined by the formulas
f* = 0^(77, 7)), 1)0),
where o> and o>° have been defined in §8.
While the preceding modes of transforming (77), (?}), and (f) into (770), (1)®), and
(f 0) are consistent with the transformations of these entities if they are actual
variations derived from some admissible family of curves, the following state¬
ments and their proofs are free from the necessity of setting up such an admis¬
sible family. To carry this idea through one must always understand that
(i?o), (1)0), and (f0) ar e formally defined as above in terms of (77) and (^), and are
not necessarily derived from variations.
We shall prove that the parentheses in (11.3) are invariant subject to (11.2c).
To that end suppose that V, 1)*, and (u) are given at t = tl on gy with (77) and
(u) subject to (11.2c). Let the formulas
(11.6) r*(e) - y*(P) + er
define a contra variant vector r* in the system (x) at the point xil(eu). Let
ziM — be the representation of the end conditions in the system ( z ). Let
c'(e) be the contravariant components in the system (z), of the vector r^e). As
we have seen previously, we have the identity
(11.7) Fri(x1(eu), r(e))x\\e u) = Q,<(zl(eu), a(e))$(eu) (h = 1, • • • ,r).
That the second parenthesis in (11.3) is an invariant subject to (11.2c) will follow
upon differentiating (11.7) with respect to e and setting e = 0. To that end we
need to establish the formulas
(11.8)
^ Fri[xl(eu), r(e)] = u^r,, */)
(e = 0),
(11.9)
± QAzKeu), <r(c)] « «•,(,* D.)
(e = 0),
where i and ifi are given by (11.4) and (11.5) in terms of ij* and ij{.
130
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[v;
To establish (11.8) we note that
4
(e = 0).
Use of (11.6) and (11.2c) then shows that
d
de Fri = Fr'rii)’ + Fr<xiV’ = u^v’
as desired.
To establish (11.9) recall that at the point z* = za(eu )
(11.10)
O’*
dz*
dx’
r'(e)
by definition of a'(e). Upon differentiating (11.10) with respect to e and setting
e — 0 we find that
(11.11)
da * _ dz{ dr’ dV dxpl
de dx1 de T dxjdxp de
(e = 0, t = tf1).
But in (11.11) when e = 0 and £ = £l,
(11.12) ^ (eu) = zgV = n”,
since (tj) is subject to (11.2c). Upon making use of (11.6), (11.11) takes the
form
(11.13)
(e = 0, t = J1)
where until after the differentiation the parenthesis is taken at the point t on g .
Referring to (11.5) we thus see that when e — 0
(11.14)
d,ji ~ **
Tt ~ Vo’
a formula of use in the proof of (11.9).
To complete the proof of (11.9) we note that
dz{ ___ dz{ dxp
de dxp de 1
and upon referring to (11.12) and (11.4) we see that
(11.15) ~ = vl (e = 0).
Formula (11.9) now follows from (11.14) and (11,15)
[12]
THE NON-TANGENCY CONDITION
131
To establish the invariance of the second parenthesis in (11.3) we differentiate
(11.7) with respect to e and set e = 0. Formulae (11.8) and (11.9) lead us to
f l and f [ o respectively, and we find that when e = 0
+ KAW = fl-o z'k + Ql<zhkuk
as desired.
The invariance of the first parenthesis in (11.3) subject to (11.2c) follows
similarly.
By a solution of the accessory problem in tensor form we mean a contravariant
vector \i defined at each point t of g and possessing components of class C 2 in
each local coordinate system into which g enters. The components of p. in a
given coordinate system shall satisfy the conditions (11.2a) and (11.2b) corre¬
sponding to this system. The components of n in any two systems covering the
neighborhoods of the respective end points of g must satisfy the conditions
(11.2c) and (11. 2d), corresponding to these coordinate systems.
We observe that the operators
Hi(i 7, X) = Li(ri) + X?7? (i = 1, • • • , m)
have the property already established for Li(rj) in §8, that the relation
yWiiv, X) = 0
is an identity in t for every set (rj) of class C2. Proceeding as in the treatment of
the equations Li(rj) = 0 in §8 we can now show that if Fx 9^ 0 along g, (11.2a)
and (11.2b) can be put in the form
iji = Ml(t, 7), X) (i = 1, • • • ,'m)
where M 1 is linear and homogeneous in the variables (77) and (1)), with coefficients
which are of class C1 in t and X.
A. W. Tucker [2] has taken up the question of the invariance of the left number
of (11. 2d) from a more general point of view. He has introduced a process of
generalized covariant differentiation appropriate to the problem and has given a
new and elegant proof of the invariance in question.
The non-tangency condition
12. We here introduce the analogue of the non-tangency condition of §6, Ch.
II. We note its invariant character and find an adequate mode of representing
it. We shall make important use of it.
The set of points in the space of the 2m variables
(a:11, • • • , xml, xl2y • * • , xm2)
which are given by the equations
xil = y *(£), xi2 = T'Xr),
132
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
for t and r near tl and t2 respectively define a regular 2-manifold which we term
the extremal manifold. The r-dimensional manifold xi9 = xi9(a) will be called
the terminal manifold. The terminal manifold and the extremal manifold inter¬
sect in the point determined by (a) = (0).
We shall assume that the terminal manifold is regular (r > 0).
Our non-tang ency condition (r > 0) is the condition that the terminal and extremal
manifolds have no common tangent line. In case r = 0 we understand that the non¬
tang ency condition is always fufilled.
One readily sees that a necessary and sufficient condition that the non-
tangency condition hold is that the matrix
(12.0)
^‘(0)
W 0
4*(0)
0 7 '(«*)
(* = 1, • • • , m; h = 1, • ■ • , r)
of r + 2 columns and 2m rows be of rank r + 2.
We consider the class of variations locally of the form
(12.1)
v * =
where p(t) is a function of class C2 in t, and y*(t) represents g. We call these
variations tangential variations . We shall show that the tangential variations
are solutions of the equations Li(rj) = 0. In fact for values of a constant e
sufficiently near zero the functions
X* = 7 Kt + ep(t))
afford admissible representations of g, and must accordingly satisfy the Euler
equations
(12.2) | Fri - F* = 0.
Upon differentiating (12.2) with respect to e and setting e = 0, one finds that (17)
in (12.1) satisfies the equations L<(i?) = 0 as stated.
These tangential variations are also solutions of the conditions (11.2b),
namely
Hi(v, X) = Lib) + A n? = 0.
In fact for a tangential variation ( rj ) the corresponding vector rj* is null as one
can verify from (11.1). In order that a tangential variation be a solution of
(11.2a), that is, ijT s 0, it is necessary and sufficient that p(t) ss 0. Thus tan¬
gential variations of the form
17* = (at + 6)t*(0 (a, b constant)
Are solutions of (11.2a) and (11.2b), and these are the only solutions of the
form (12.1).
When we come to conditions (11.2c) we have the following lemma.
[13]
CHARACTERISTIC SOLUTIONS IN TENSOR FORM
133
Lemma 12.1. If the non-tangency condition holds , there are no non-null tangential
variations which are solutions of conditions (11.2a) and (11.2c).
If the lemma were false, there would exist constants a and b not both zero,
and constants (u), in case r > 0, such that
(at1 + - xi'u*' - 0,
(at2 + 6)W) - = 0.
The matrix (12.0) could not then be of rank r + 2. From this contradiction we
infer the truth of the lemma.
Characteristic solutions in tensor form
13. Relative to our accessory problem in tensor form we formally define char¬
acteristic solutions, characteristic roots, and indices of characteristic roots as in
Ch. II, §4. Characteristic solutions are defined by contra variant vectors while
characteristic roots and their indices are invariants.
If one refers the neighborhood of g to the normal coordinates of §1 and sets
up the corresponding non-parametric problem as in §5, one thereby obtains a
special non-parametric accessory problem which we shall term a normal accessory
problem in non-parametric form. Concerning this accessory problem all the
results of the non-parametric theory are available. The principal object of this
section will be to relate the general accessory problem in tensor form to this
normal accessory problem in non-parametric form.
We represent the neighborhood of g in terms of the normal coordinates of §1.
As in §5 we then set
(Vl, ■ ■ ■ , Vn, x) = {x\ ■■■ ,xm)
and
(13.1) /Or, 2/1, * • - , Vn, Pu • * * , Vn) = F(yl} • • * , yni ph * • • , pn, 1).
Corresponding to the x axis as an extremal we set up the form 8(i?, rj') as in Ch.
II except that it will be convenient here to use superscripts on the n = m — 1
variables tjm, instead of subscripts. In terms of the given end conditions
x" = x"(a) (i = 1, • • • , m),
the end conditions in the non-parametric form become
y\ = = **“(<*) (m = i, •••,»).
(13.2)
X» = X*(a ) = x**(a).
The accessory problem of Ch. II now becomes our normal accessory problem
in non-parametric form. It is given as follows :
f- - Q*. + Xij' = 0
(13.3a)
(/» = 1, • , n),
134
THE FUNCTIONAL ON A RIEMANNIAN SPACE [ V ]
(13.3b) »'* - yUu‘ * = 0 [(<*) = (0); s = 1, 2],
(13.3c) yUC1 ~ VUV + bkkU“ = 0 (h, Jc = 1, • ■ ■ , r),
with
(13.4) hhk =6^ + [/*£*+ f.xlxl + fv(,x'hy’ltk + x‘kyU) + ]’>
where we have added a star to f *(x) to distinguish it from f »•(£) in the parametric
form.
The variations 17 '‘(x) in the non-parametric problem will be distinguished
from the variations i?*( t ) in the parametric problem by the use of the super**
scripts n. We understand that jjl = 1, ••• , n — m — 1 and i = 1, • • • , m.
We turn to the accessory problem (11.2) in tensor form. If the coordinates
are normal, the components of (?j) tangent and orthogonal to g are given by the
equations
VT = 77™,
(13.5) T)l = 77* (m = 1, • • • , n = m - 1),
vZ= 0,
as follows from (11.0) and (11.1). Moreover the last equation in (11.2b) here
takes the form
Lmiv) + Xl7m = Lm(v) 25 0,
and may be discarded. See (13.8)" and (13.9).
The problem (11.2) then becomes what we term the normal accessory 'problem
in parametric form. It is as follows:
(13.6a)
vm = o,
(13.6b)
Ln(y) + Xij" = 0
(m=1,--
■ , n),
(13.6c)
Tp* — x'k‘uk = 0
(t = 1, • • • , m; s =
1,2),
(13.6d)
\ + PhkUk = 0
(h, k = 1, ■ ■
• , r).
We shall show that the problem (13.3) is essentially equivalent to the problem
(13.6). Before coming to the principal lemma we need to evaluate the partial
derivatives of F in terms of those of /.
From the definition of/ in (13.1) and the homogeneity of F we have
(13.7) rmf(x, = F(yu • • • , yn, x, r\ ■ ■ ■ , rm) (r“ > 0).
We see then that along the x axis (r1 = • • • = rn = 0, rm = 1)
(13.8) ' Fr» = /„, F Tm = /,
(13.8) Fxnrv = fyppr, Ftnrm = fyp (/*» V = 1, * * • , n)
[13]
and
CHARACTERISTIC SOLUTIONS IN TENSOR FORM
135
(13.9)
JP^Pp
0
0
0
(h j = 1, • ■ • , rn).
From (13.8) and the fact that the x axis is an extremal, we find that along the
x axis
(13.10) F r»xm = fyj Frmxm — fx (m = 1) ‘ * ) U) ,
We shall now prove the following lemma.
Lemma 13.1. If y^t), i = 1, • • * , m, satisfies (13.6) with constants X and (u)}
the corresponding functions y*(x), y = 1, • • • , n, satisfy (13.3) with the same
constants X and (?/).
To show that the functions y^{x) satisfy (13.3a) we observe that any function
rjm(t) of class C2 defines a tangential variation (0, • ■ • , 0, ym) and satisfies the
conditions
Li{ 0, • * • , 0, ym) s 0 (i = 1, • * • , m).
Hence we have identically,
* * * , Vm) s Li(yl9 • ■ ■ , yn, 0).
But from (13.8)" and (13.9) we see that if t = x,
i ^ 0) “ ^ “ fy* (m = 1, * • * , n).
Combining these two identities we find that
Lfiiy) + Xr;M = — — 12^ -+- X77m (m == 1 , - n)
provided / = x. Thus the lemma is true in so far as the satisfaction of conditions
(13.3a) is concerned.
Moreover the conditions (13.3b) are a consequence of conditions (13.6c), in
fact are a subset of conditions (13.6c). We continue with the following:
(A). The conditions (13.3c) are satisfied by y^ix), the constants (u), and cor¬
responding functions
To prove (A) we shall evaluate the various entities entering in (13.6d) in
terms of entities entering in (13.3c), substitute our results in (13. 6d) and thereby
obtain (13.3c).
From the definitions of bhk and fihk we find that
- [ftXlz'k + fyphVU +
(13.11)
fihk — bhk
136
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
making use thereby of (13.8). Making use of (13.9) we see that
f* = f* + (m = 1, • • • , n),
~ F rms%Vi (t = 1, * , Tit),
and then upon using (13.10), (13.8) and (13.6c) we find that (for s not summed)
r; = tv + /y/iu* (m « 1, * • • > n),
(13.12)
(fvy>k +
Upon substituting the right members of (13.11) and (13.12) in (13.6d), (13.3c)
results as stated.
The lemma is thereby proved.
The preceding lemma will be strengthened and completed in the following
theorem.
Theorem 13.1. If the non-tangency condition holds , then for a given X there is a
one-to-one correspondence between the solutions of the normal accessory problem in
parametric form and the solutions of the normal accessory problem in non-para-
metric form in which a solution
(13.13) r = Sit) (i = 1, ••• ,m)
of (13.6) corresponds to the solution
(13.14) ^ = <p*(x) (m = 1, ■ • * , n)
of (13.3). Moreover under this correspondence linearly independent solutions
correspond to linearly independent solutions.
A solution of (13.6) uniquely determines the constants ( u ) with which it
satisfies the terminal conditions (13.6c) since the terminal manifold by hypoth¬
esis is regular. According to the preceding lemma a solution v?‘(0 of (13.6)
with its constant X and above constants (w) will determine a solution (13.14)
which will satisfy (13.3) with the same constants (u) and X.
On the other hand two solutions (13.13) which determine the same solution
(13.14) must be identical. For their difference would be a solution of (13.6)
of the form
(o, • • • , o, A *>»«))
and thus a tangential solution. But we have seen in §12 that if the non-tangency
condition holds, tangential solutions of the accessory problem in tensor form must
be null. Thus one and only one solution of the form (13.13) gives rise to the
solution (13.14).
Finally each solution (13.14) of (13.3) gives rise to a solution (13.13) of (13.6).
[14]
THE GENERAL INDEX FORM
137
For if the solution (13.14) satisfies (13.3) with constants X and (u), the functions
^*(0, of which rjm = cpm(t) satisfies the conditions
Vm - 0, 7]ma = xl(0 )u* (* - 1, , r),
will satisfy (13.6).
The preceding shows that the null solution corresponds to the null solution,
and from this it follows that linearly independent solutions correspond to linearly
independent solutions.
The theorem is thereby proved.
The general index form
14. We now suppose the Legendre ^-condition of §7 holds along g , and that g
satisfies the transversality conditions. We are also assuming that the terminal
manifold is regular and that the non-tangency condition holds.
Suppose the arc length t on g increases from tl to t2 inclusive. Let
a o9 y dpi dp+i (o o dp-\i
be a set of increasing values of t so chosen as to divide g into segments on which
there are no pairs of conjugate points. We cut across g at the point at which
t = aqy q = 1, • • * , p, by a regular 7wiimensional manifold Mq of class C2, of the
form
a*1' = Xi(fih ■ • • , dn) (n = m - 1),
intersecting g when (/3) = (0), but not tangent to g . We term the manifolds
Mq intermediate manifolds. Let
(14.1) A1, P\
be a sequence of points of which A1 and A2 are admissible end points determined
by parameters (a) in the end conditions, and Pq is on the manifold M q near g.
Points (14.1) sufficiently near g can be successively joined by extremal arcs
near g to form a broken extremal E 0. Let (r) be a set of parameters of which
the first r are the parameters (a), and the remaining the successive sets of param¬
eters (0) of the points Pq . The value of J along the broken extremal E0 will
be denoted by J(v). The function J(v) will be termed an index function belong¬
ing to g, to the given functional, and to the end conditions.
Our basic index form is the form
P(z) = JViVj(0)ZiZi ( hi ~ 1, ••■ , *)
where 6 is the number of variables ( v ).
The index form P(z) is an invariant clearly independent of the local representa¬
tions of R used to evaluate it. If in particular we represent the neighborhood of
i i by means of a normal system of coordinates y)> the index form P(z) may be
identified with the index form Q(z, 0) of Ch. II set up for the segment
tl g x £ t2
THE FUNCTIONAL ON A RIEMANNIAN SPACE
138
[V]
of the x axis as an extremal. We must of course use the same intermediate mani¬
folds and the same variables (v) in both cases.
By virtue of the correspondence between characteristic solutions of the acces¬
sory problem in tensor form and characteristic solutions of the normal accessory
problem in non-parametric form as given in Theorem 13.1, and by virtue of the
results of Theorems 2.2 and 2.3 of Ch. Ill concerning Q(z, 0), we have the
following fundamental theorem.
Theorem 14.1. The nullity of the index form P(z) equals the index of\ = 0 as a
characteristic root of the accessory problem in tensor form, and the index of P(z)
equals the number of characteristic roots of this problem which are negative.
The following is an easy corollary of the theorem and the relation of its condi¬
tions to the corresponding conditions in non-parametric form.
Corollary. In order that an extremal g afford a proper , strong, relative minimum
to J under our general end conditions, it is sufficient that g satisfy the transversality
conditions, that the Weierstrass S-condition },old along g, that Fi 0 along g, that
the non-tangency condition hold, and that all characteristic roots of the accessory
problem be positive.
Particular consequences of the hypotheses of the corollary of importance in
its proof, are that the Legendre ^-condition holds, that P(z) is positive definite,
and then from the non-parametric theory, the fact that there will be no pairs of
mutual conjugate points on g. The conclusion of Lemma 7.1 also holds and the
proof can be completed with a suitable use of J(v) and Mayer fields.
The case of end manifolds
15. We shall now take up the question of the existence of a family of extremals
cut transversally by a manifold M. As far as the author knows this has not
previously been treated for the case of general dimensionality.
Let M be locally represented in the form
(15.1) x * = <p*(al, • • • , ar) (0 < r < m)f
where the functions are of class C3 for (a) near (a0). We suppose that M is
regular, and cuts g transversally when (a) = (a0), at g’ s first end point. In
the functional J we suppose 6(a) is of class C3. We begin by seeking solutions
(a) and (r) of the transversality conditions
(15.2) ' FrM«), rW(a) + 6h(a) = 0 (A = 1, • • • , r)
and the side condition
(15.2) " w = 1.
Here h indicates differentiation with respect to ah.
We shall suppose that g is regularly represented by functions x < = xl{t) with
t = t o at the initial point of g . We suppose that t is the ordinary arc length of g
[15]
THE CASE OF END MANIFOLDS
139
in the euclidean space ( x ). We denote the values of (x) and (x) on g when
t = to by (x0) and (r0). Our initial solution of (15.2) is then (a, r) = (a0, r0).
Let a* (a), k = 1, • • • , m — r, be a set of m — r independent solutions of the
homogeneous equations
<n<Ph(.<*) = 0 (h = 1, ■ • * , r).
These soluti6ns can be so chosen as to be of class C2 in (a) for (a) near (a0)*
Conditions (15.2) can then be written in the form
(15.3) ' FM*)> r) + pM(«) = Ai(a),
(15.3) " r'r* = 1,
where A*(<x) is a particular solution of class C2 of the equations
A up l (a) + dh(a) = 0.
The variables (p) must now be added to our unknowns. Let (p0) represent the
set (p) which satisfies (15.3) with (a, r) = (ac, r0).
The matrix of the partial derivatives of the left members of (15 3) with
respect to (r) and (p) has the form
F • ■
1 r'rl
k
2 r’
0
(hj = L * ' , = 1, • • • ,m - r).
Now at least one of its (m + 1) -square determinants A* obtained by omitting
all but the fcth of the Iasi m -- r columns does not vanish at {a0, r0). For we
have
A* = -2 x, r)r'o\
by virtue of (2.11). Thus A* = 0 for all values of k only if in the euclidean space
(x) the (m — r) directions (or/;) are orthogonal to the direction (r0). But the
directions ( <rk ) are orthogonal to M at (a0) and constitute a base for such direc¬
tions. Any direction orthogonal to all of the directions (ak) must be tangent to
M at (q0). Hence if A* were zero for each value of fc, the direction (r0) would be
tangent to M at (a0) contrary to the non-t angency condition. Thus at least one
of the determinants A*, say ATO_r, is not zero.
The equations (15.3) can accordingly be solved for p™_r and the variables (r)
in terms of the variables (a) and the variables p i, • • • , pm_r_i, at least neighbor¬
ing the initial solution (ao, r0, po)* Let us set
(15.4)' vk = pk} vk0 = pifco (k = 1, • ■ * , q = m - r - 1),
and write the solution in the form
(15.4)" r{ = r*(a, v),
Pm—r — Pm—ri&y
140
THE FUNCTIONAL ON A RIEMANNIAN SPACE
[V]
for (a, v) near (a0l Vo). Following the methods of §4, taking <p as riri} we can now
be assured of the existence of an ( m — 1) -parameter family of extremals of the
form
(15.5) x 1 = hx(ty a} v)
along which t is the arc length in the space (x), and which satisfy the initial
conditions
(15.6) <p'(a) = a, v)y
(15.7) r'(a, v) s h)(t0, a, v),
where hx is of class C2 in its arguments near (/0, a0, v0). The extremal of this
family determined by (a, v) will be cut transversally by M when t — t0.
We shall now establish the following theorem.
Theorem 15.1. I1 he family of extremals (15.5) cutting M transversally are so
represented that the Jacobian
M{t) «
, *m)
id{t , * ) airy Viy * y v ,f)
l(«, »’) = («o, t'o) ; r + q = m - 1],
evaluated on g vanishes at t — t0 to the qth order.
Without loss of generality we can suppose a non-singular linear transformation
of the variables (x) has been made so that
(15.8) rj = • • •
and on g at the initial point of gy
(15.9)
rTl = 0,
I 0
0 0
where 7 is a unit (m — 1) -square matrix.
By virtue of (15.6) we see that the last q columns of M(t) vanish at <0. We
accordingly have
M(t) = (t- t0)9 A (t)
where a use of (15.6) and (15.7) discloses the fact that
(15.10) A(t0) = |ri,^(a0)lrJik(ao) |.
Here h =■ 1, • * ■ , r, k = 1, * • • , q} and A(t) is continuous in t for t near t0. We
shall show that i4(20) s* 0.
To that end we regard (15.3)' and (15.3)" as identities in (a, v) subject to
(15.4). Upon then differentiating (15.3)" with respect to Vh and using (15.8)
we see that
r?A(<*o, Vo) = 0
(h = 1, ■■■ , q).
[15]
THE CASE OF END MANIFOLDS
141
Thus the last q columns of A (to) are orthogonal to the first. Upon similarly
differentiating (15.3)' and using (15.9) we find at (a0y v0) that
(15.11) < + a) + P7“ VTr = 0 (h = 1, • • • , q).
From (15.11) we see that the last q columns of A (to) are orthogonal to M at (<*0)
and are moreover independent directions. In sum the last q columns of A(t0)
represent directions orthogonal to the first r + 1 columns of A(t0). Since the
first r + 1 columns are likewise independent, A(U) ^ 0.
The proof of the theorem is now complete.
We can, if we please, change the parameter t in the family (15.5) to the arc
length on R . With this understood we can “continue” the family (15.5) as in
§4. The resulting jacobians of the form of M(t) will be called the focal deter¬
minants corresponding to the manifold M . We term their zeros on g the focal
points of M. Exactly as in the case of the determinants defining conjugate
points in §5, so here, we can introduce normal coordinates and show that the
focal determinants vanish at the same points and to the same orders as the focal
determinant of M in the non-parametric theory. A first conclusion is that if g
affords a weak minimum to J, it is necessary that there be no focal point of M
between the end points of g.
The basic theorem here is the following :
Theorem 15.2. If M cuts g transversally at g’s initial point A' without being
tangent to g at A1, and if the Legendre S-condition holds along gy the index form P(z)
corresponding to the conditions that A1 lie on M and A2 be fixed has an index equal to
the number of focal points of M on g between A 1 and A2, The nullity of P(z) equals
the index of A2 as a focal point of M.
With the aid of this theorem one sees that sufficient conditions that an ex¬
tremal afford a proper, strong minimum to relative to admissible curves which
join the manifold M to the second end point of gy are that M cut g transversally
without being tangent to gy that Fx ^ 0 along gy that the Weierstrass SKJondition
hold along gy and that there be no focal point of M on g between M and A 1
including A2.
The final theorems on the case of two end manifolds as given in Ch. Ill can be
similarly carried over into theorems valid on R. The same is true of the the¬
orems on periodic extremals to which we shall return in Ch. VIII. In general
the results of this chapter furnish a mechanism which enables one to pass
freely from the parametric to the non-parametric case. The results are freed
from the necessity of holding to a single euclidean space or any one coordinate
system, and, most important of all, the invariant or tensor forms of the basic
elements and hypotheses have been set forth.
CHAPTER VI
THE CRITICAL SETS OF FUNCTIONS
The theory of critical points of functions is concerned with the relations of
critical points, classified in the small, with the topological characteristics of the
domain on which the functions are defined. The basic relations were first
discovered for the case of non-degenerate critical points, that is, for critical points
at which the hessian of the function is not zero. To extend the theory one
met the difficult and basic problem of characterizing degenerate critical loci so
that these loci might be counted as finite sets of non-degenerate critical points.
Such an extension led to a radical change in the topological aspects of the theory.
Deformations entered more, and combinatorial analysis situs less.
The choice of methods has been largely influenced by the desire to adopt a
procedure which might serve as a model for the case of functionals. It has
been found that the underlying theory can be given a relatively abstract topologi¬
cal form of great elasticity. This abstract form embraces three different par¬
ticularized theories, namely, the theory of critical points of the present chapter,
the theory of functionals of the following chapter in which the curve replaces
the point, and the theory of the space ft of Ch. VIII in which subgroups of sub¬
stitutions play so large a part. Each of these three theories remains highly
individual in the nature of the deformations peculiar to it.
The present chapter contains a number of applications. It is impossible
however to give here an idea of the scope of the theory from this point of view.
It will be sufficient to say that such applications are numerous in analysis,
geometry, and physics, and the number is constantly increasing (Kiang [1, 2, 3],
Birkhoff [7]).
The non-degenerate case
1. Let / be a single-valued function of a point on a circle. Suppose that / is
of class C2 in terms of the arc length on the circle. Suppose also that /" 0
when/' — 0. Let M0 and M i be respectively the number of relative minima
and maxima off. We have the relations
Mo > 1,
M0 - ilfi = 0.
To proceed directly to a general case suppose that / is a single-valued function
of the point P on the Riemannian manifold R of Ch. V. We suppose that / is
not constant on R. In terms of each set of local coordinates (x) we also suppose
that/ is a function >p{x) of class C2. We term such a function/ admissible.
A point on R at which each of the first partial derivatives of \p(x) vanishes will
142
[1]
THE NON-DEGENERATE CASE
143
be called a critical point of /. Suppose that (x) = (0) defines such a critical
point in the system ( x ). If
I ’/w(O) I ^ 0 O', j =1 , • • ■ , m),
the critical point (x) = (0) will be termed non-degenerate. One sees that the
property of non-degeneracy of a critical point is independent of the local co¬
ordinate system (x) employed to represent /. If the critical points of / are all
non-degenerate, / will be termed non-degenerate. In case / is non-degenerate one
recognizes that the conditions
tx* = 0 O' = 1, * • • , rn)
have at most isolated solutions. The critical points of / on R are then isolated,
and hence finite in number.
As is well known, a suitable, non-singular, homogeneous, linear transformation
of the variables (:r) into a set of variables ( z ) will effect a reduction
— — z\ — * ' - — z\ + z\jrl + • • * + z2r,
where 0 g k :g r. Here r — m if the critical point is non-degenerate. The
number k is called the index of the critical point . It is clearly independent of the
local coordinate system used to represent /. There are m + 1 possible indices
for a critical point. A non-degenerate critical point of index zero affords a
relative minimum to/, while one of index rn affords a relative maximum.
The following theorem comes first in the history of our development of the
subject (Morse [1], Morse [20] with van Schaack).
Theorem 1.1. The numbers M\ of critical points of index i of a non-degenerate
function f defined on R, and the connectivities Rj (mod 2) of R, satisfy the following
relations :
Mq ^ R()
Mo - M1 ^ Ro - Ri,
(1.1) Mo - M, + M2 ^ Ro - Ri + ft,
Mo - Mx + . • • +(-ir Mm - Ro - ft + • • +(~1 )mRm.
A proof of this theorem will be a part of a treatment of the general case which
includes the degenerate as well as the non-degenerate case.
Of the relations (1.1) the first is merely a statement of the necessity of the
existence of at least Ro relative minima. The second relation in the form
Mi ^ Mo T" — Ro
is essentially BirkhofTs minimax principle (Birkhoff [1]) although not stated by
Birkhoff in precisely this form. The last relation in the case m == 2 was known
to Poincar6 [1]. The last of these relations for the general m was discovered
144
THE CRITICAL SETS OF FUNCTIONS
[VI]
independently by the author at about the time Lefschetz [3] and Hopf [1] proved
the corresponding basic equality concerning the signed index sum of fixed points
of a transformation. In this connection we note the following corollary of the
theorem.
Corollary 1.1. The numbers Mi and R% of the theorem satisfy the relations
(1.2) Mi ^ Ri (i - 0, 1, • ■ • , m).
From the set of all relations (1.1) one can thus infer the existence of at least
R 0 “f“ R 1 ~f~ ' * ’ + Rm
critical points on R.
We regard the above corollary as a statement of the number of critical points
which are topologically necessary . We term the number
Qt = Mi - Ri
the number of critical points in excess of those topologically necessary. The
relations (1.1) imply much more than the relations (1.2). In fact they imply
the relations (1.2) together with the necessary limitations on the numbers Qt.
As a particular example of such a limitation we state the following corollary.
Corollary 1.2. hi the non-degenerate case the numbers Q% satisfy the relations
Qi~ i + Qt+ 1 ^ Qv (i = 1, • • • , m - 1).
These relations follow from (1.1) upon comparing each relation with the third
following relation.
In particular if R is an fw-sphere, m > 1, we have the relations
+ Mm ^ Mi (i = 2, • • • , m - 2),
together with the special relations
Mq T M 2 ^ M\ T 1,
Mm + Mm~ 2 ~ Mm- 1 + 1.
Many other conditions on the numbers Qi can be derived from the relations (1.1).
We shall now indicate certain extensions of Theorem 1.1 which we shall not use
in these Lectures, and shall accordingly not establish. These extensions are of
importance in connection with the question of the completeness of the relations
(L1)*
Suppose that the domain of definition of f is the interior and boundary of a
region 2 of R. Suppose the points on the boundary B of 2 neighboring any one
such point satisfy a relation of the form
F(xl, • • • , xm) = 0
[ 1 j THE NON-DEGENERATE CASE 145
in terms of the local coordinates (x), where F(x) is of class C3, and
FxiFxi 7*0 (i = ], • • - , m).
We term 2 a regular region. We state the following theorem.
Theorem 1.2. Iff is non-degenerate on 2 and on the boundary B of 2 possesses a
positive directional derivative fn along the exterior normal , then the numbers Mi of
critical points of index i of f on 2 and the connectivities Rt of 2 again satisfy the
relations (1.1).
When the proof of Theorem 1.1 has been completed the reader will be able to
construct a proof of Theorem 1.2 upon reading the last section of Morse [1].
We remark that Theorem 1.2 also holds if 2 is a bounded region in euclidean
ra-space. That the relations between the integers Mi and Ri are the only
relations which always hold between these integers alone follows from the
following theorem.
Theorem 1.3. Corresponding to any prescribed set of integers Miy Riy i = 0,
• • ■ , my positive or zero , satisfying the relations (1.11 with M0 and R0 positivey there
exists a regular region 2, together with a non-degenerate function /, defined on 2 and
assuming an absolute non-critical maximum on the boundary of 2, such that the
integers A\ are the connectivities of 2 and the integers Mi are the numbers of critical
points of f of index z.
While assured of the truth of Theorem 1.3 the author has never published a
proof. The theorem is stated in Morse [11]. An independent proof of the
theorem has been given by a pupil of Professor Courant, Dr. John. See John [1].
We can extend Theorem 1.2 still further by removing the condition /„ > 0 on
the boundary B. On B let / equal a function L . As a function of the point on
By L will have its own critical points with their indices. Instead of the assump¬
tion /„ > 0 on B we now assume merely that/ has no critical points on By and
that the function L is non-degenerate as a function of the point on B. These
conditions will in general be fulfilled. We term them the general boundary
conditions onf. The theorem is as follows.
Theorem 1.4. Under the general boundary conditions on f the relations (1.1)
still holdy where Ri is the ith connectivity of 2 and Mi is the number of critical points
of index iy not only of f on 2 but also of L at points on B at which fn < 0.
For a proof of this theorem in euclidean n-space see Morse and Van Schaack
(Morse [20]) . The proof in general is similar.
W. M. Whyburn [1] has developed certain interesting aspects of the theory of
critical points of functions in the case where the critical values are not necessarily
isolated.
The equality in the relations (1.1) in the case of a simply connected region in
n-space can be derived with the aid of the theory of the Kronecker characteris¬
tics, although Kronecker apparently made no such explicit derivation. See
Kronecker [1, 2].
146
THE CRITICAL SETS OF FUNCTIONS
[VI]
The problem of equivalence
2. Before coming to the problem of equivalence we shall enumerate certain
conventions concerning singular chains on R. See Lefschetz [1, 2]. We shall
vary the form of the basic definitions slightly, in a way that makes the work of
the present chapter capable of a natural generalization in later chapters. We
wish here to acknowledge the benefit derived from an interchange of views with
Dr. A. W. Tucker on the various means of defining singular chains and cycles.
See also Alexandroff [1], Alexander [1, 2], Tucker [1].
Let ak and ft be two Avsimplices in a euclidean space En. A non-singular,
affine, projective correspondence between ak and ft will be termed an affine
correspondence between ak and ft. If ak lies in a euclidean space /?„, and ft in a
euclidean space Em w ith n ^ m, we identify En with the linear subspace of Em
determined by the first n coordinate axes of Em, and define an affine correspond¬
ence betw een ak and ft as before.
Indicating closures by adding bars, let <p represent a continuous map of at,
on R. The image ak of ak under (p will be termed a k-cell on R. Let bk be a
second fc-cell on R defined with the aid of a map \p of ft on R. Let T be an affine
correspondence between ak and ft. If points on ak and ft which correspond
under T possess the same image on R under <p and \[s respectively, the cells ak
and bk will be regarded as identical on R. We shall refer to this statement as the
convention of identity.
If cti ks any z-sirnplex on the boundary of a* , the image of a, under <p will be
said to be a boundary i-cell of ak on R. The boundary of ak on R is however still
to be defined.
We shall deal only with unoriented cells, and with cells mod 2.
By a closed z-cell on R wre mean an z-cell on R together with its boundary j-cells.
By an i-chain on R we mean a finite set (possibly null) of closed z-cells on R}
no two of which are “identical.” By the sum mod 2,
Zi + Wi (mod 2),
of tw7o t-chains z* and u\- on R, we mean the set of closed z-cells which belong to
Zi or iv i but not to both Zi and u\-.
Let k and r be integers with r < k. Let ak and br be cells on R given as
continuous images of simplices ak and ft. Suppose ft is the affine projective
image of ak under a singular transformation T in which each point of ak corre¬
sponds to a unique point of ft, and each point of fa corresponds to at least one
point of ak. If points which correspond on ak and ft possess the same images on
R on the cells ak and br, then ak will be termed “degenerate.” Cf. Lefschetz [2].
Degenerate A;-cells wdll be counted as if null in any A;-chain on R,
The boundary z*_i of an 2-chain Zi on R is defined as the sum mod 2 of the
closed (i — l)-cells which are the boundary cells of z-cells of z,. One then writes
(2*1) Zi — * Zi_! (mod 2).
We observe that the boundary of the sum of a set of z-chains is the sum of the
[2]
THE PROBLEM OF EQUIVALENCE
147
boundaries of the respective chains. It appears that bounding relations such as
(2.1) can be added by adding the respective members of the bounding relations,
mod 2.
A chain a, on R whose boundary is null is termed a j-cycle. A /-cycle will be
said to be bounding or homologous to zero if a, is the boundary of some (j + 1)-
chain aJ +i on R. One then writes
(ij ~ 0 (on R ).
This is understood, mod 2. This phrase will ordinarily be omitted. One sees
from the way bounding relations can be added, that homologies
a, ~ 0, bj ~ 0 (on R)
imply
a,} + 6/^0 * (on R ).
The last relation will also be written in the form
dj ~ bj.
With this understood it appears that valid homologies can be combined into a
valid homology by adding the respective members mod 2.
By a proper linear combination of a finite set of ft-cycles is meant a linear
combination of these cycles with coefficients which are not all zero mod 2. By a
proper homology between a set of /r-cycles is meant an homology
X ^ 0
in which X is a proper linear combination of cycles of the set. A set of /c- cycles
on R will be termed independent on a domain A if no proper linear combination
of these cycles bounds on A .
Let a class C of fc-cycles be distinguished by the possession of certain properties
B. By a maximal set of cycles of C will be meant a set of cycles of C, every
proper linear combination of whose cycles belongs to C and which contains the
maximum number of cycles of C of any set with this property. As a convention
we admit the possibility that the number of cycles in a maximal set may be
infinite.
To subdivide a /-chain a, on /£, we subdivide the simplices representing its
respective cells, and take the resulting images of the new simplices as the new
cells. If two simplices a* and correspond under an affine collineation T by
virtue of which their images a» and bi are identical on R, the simplices a* and ft
shall be subdivided so that the subdivision of ft- may be obtained from that of
a* by applying T. This is clearly possible at least for those modes of subdivision
which subdivide cells in the order of dimensionality.
We return to the function/ on R. We no longer assume that the critical points
are non-degenerate. We shall assume however that the number of critical values
THE CRITICAL SETS OF FUNCTIONS
148
[VI]
of / is finite. This assumption is always fulfilled in the analytic case. We
suppose that / is of class C2 and not identically constant.
By a critical set a will be understood any closed set of critical points on which
/ is a constant c, and which is at a positive distance from other critical points of/.
A critical set may or may not be connected (in the point set sense), or be a finite
complex. In the analytic case the critical sets are at most finite in number, with
dimensionalities varying from 0 to m — 1 inclusive. If a contains all the critical
points at which / = r, it will be called a complete critical set corresponding to c.
In the analytic case a complete critical set is composed of a finite ensemble of
connected critical sets.
Since the non-degenerate case occurs in general, and since the relations (1.1)
give a complete set of conditions on the existence of non-degenerate critical points,
it is natural to seek to assign to each critical set <r an ideal “equivalent/’ set G
of non-degenerate critical points in such a fashion that the relations (1.1) still
hold. But the property that the relations (1.1) still hold is only one of the
properties that we shall require of this equivalent set G. The problem of
equivalence is the problem of specifying the properties which the set G should
have in order that it may fairly deserve the name of a set equivalent to a. This
question of equivalence arises in algebraic geometry, for example, when the
geometer asks how many double points a multiple point shall be equivalent to,
or in the case of fixed points of transformations, when the geometer seeks to
count complicated loci of fixed points as equivalent to a finite set of fixed points
of simple type.
We shall begin with the case of a complete critical set a corresponding to a
critical value c. Let a and h be any two constants w hich are not critical values of
/, which are such that a < c < b, and such that c is the only critical value of /
between a and b. If c is the absolute minimum of /, the domain / < a is vacuous.
We shall give a definition of an ideal set of non-degenerate critical points
equivalent to the complete critical set a. Later we shall find it possible to extend
this definition to critical sets which are not complete.
Relative to the above critical value c and the preceding constants a and b, a new
A:-cycle shall mean a A:-cycle which lies on the domain / < b but is independent on
/ < b of A;-cycles on f < a. Relative to the critical value c and the constants a
and 6, a newly-bounding Zr-cycle shall mean a A;-cycle on / < a independent on
/ < a, but bounding on f < b. It will follow from Lemmas 2.1 and 2.2 that the
numbers
mt, ml
of cycles in maximal sets of new fc-cycles and newly-bounding ( k — l)-cycles
respectively, are finite and independent of the choice of the numbers a and b
among numbers which are not critical values of /, and between which c is the only
critical value of /.
We set
* m* + m*,
12]
THE PROBLEM OF EQUIVALENCE
149
and say that the complete critical set a is equivalent to mk non-degenerate critical
points of index k. We term the integers
m o, m i, • • * , m,
the type numbers of the critical set a.
In §8 we shall justify our definition of equivalence by establishing the follow¬
ing four properties of the numbers mk.
I. If a is a set of non-degenerate critical points , the corresponding type number
mk of a equals the number of non-degenerate critical points of index k in a .
II. The numbers mk are completely determined by the definition of f in an arbi¬
trarily small neighborhood of the critical set a.
III. If each critical set a is counted as equivalent to m.k ( k — 0, • • • , m) critical
points of index k, the relations (1.1) still hold.
IV. Suppose the function f is analytic and is approximated for parameters
(t*u * • * l Mr)
near the set (0) by a function <t> of the point on R and the parameters (y) which y in
terms of the local coordinates (x) of R and of the parameters (y), is of the form F(x, y),
where F(x, y) is of class C 2 and non-degenerate for (y) ^ (0). If
for (y) = (0), then for (y) ^ (0) but sufficiently near (0), <P will possess at least mk
non-degenerate critical points of index k neighboring the given critical set a of f.
Relative to property II we remark that the numbers mk do not possess property
II except by virtue of a deep lying proof. For property II implies the invariance
of the numbers mk with respect to all functional alterations of / which leave /
invariant neighboring <r, and replace / by a function which is again admissible
on Ii. Moreover examples will show that the numbers m\ and mf do not in
general separately possess this property of functional invariance, although their
sum nik does. We here have a distinction between functional and topological
invariance. For the numbers mfk and m 7 are invariant under any homeomorph-
ism of R which preserves the value of /, but are not necessarily invariant with
respect to the above functional alterations.
Property III is fundamental in proving the existence of critical points, and
property IV interprets this result in terms of non-degenerate functions approxi¬
mating/. We shall give further point to property IV by showing that when the
critical set a lies in a single coordinate system (x) and / is analytic, an approximat¬
ing function such as $ always exists.
That the number of A%cycles in maximal sets of new or newly-bounding k-
cycles relative to c is finite follows from the following lemma.
Lemma 2.1. If a is an ordinary value of /, the connectivities of the domain f < a
are finite.
150
THE CRITICAL SETS OF FUNCTIONS
[VI]
To establish this lemma we shall make use of the trajectories orthogonal to
the manifolds/ constant, represent ing these trajectories in the form
dt rfjx,
0‘> j = 1, • • , w).
Here is the cofactor of in | gtJ | divided by | gu |. Along these trajectories
df = atr1
so that we can suppose / - t along such trajectories.
Let c be a positive constant so small that no value of / between a and a — e
inclusive is a critical value. We can deform the domain f < a onto the domain
f ^ a — e, moving each point on the domain
a — e ^ f < a
along the orthogonal trajectory through the point so that / decreases at a unit
rate with respect to the time r, stopping the movement when the point reaches
the manifold f — a — e.
Now let the domain / ^ a — e be covered by a complex Cm of cells of or a
subdivision of these cells, taking this subdivision so small that Cm lies on / < a.
Any cycle on / < a will be homologous, by virtue of the above deformation,
to a cycle on / g a — c and hence on Ctn. A maximal set of k~ cycles on / < a,
independent on/ < a, will contain at most the number of A>cycles of cells of Cm
which are independent on Cm , and this number is finite. The lemma follows
directly.
With Lemma 2.1 we naturally associate the following lemma.
Lemma 2.2. If a and b are any two ordviary values of f with no critical values
between them , the domains
f < a, f < b
are hom eo?n orphic .
To prove the lemma choose a — e as in the preceding proof. We establish a
homeomorphism between the domains f < a and / < b as follows. Let p be a
point at which / = /0 where
a - e g /0 < b.
Suppose p lies on the orthogonal trajectory X. We make the point p at which /0
divides the interval ( a — e , b) in a given ratio correspond to the point on X at
which/ divides the interval (a — e, a) in the same ratio. The remaining points
of/ < b shall correspond to themselves. The correspondence between the
domains / < a and / < b is nowr one-to-one and continuous, and the proof of the
lemma is complete.
[3]
CYCLES NEIGHBORING <x
151
Cycles neighboring a
3. In this section we suppose that there is just one critical value c between a
and b. It will be convenient to say that a point on R at which / < r is below c.
Let a be a critical set off on which/ = c. The set a may or may not be com¬
plete, that is, contain all the critical points at which / = c. By a neighborhood
N of a we mean an open set of points which includes all points of R within a small
positive geodesic distance of a. We admit only such neighborhoods of a as lie
on the domain
(3.1) a < f < b.
A neighborhood N of <r will be termed arbitrarily small if its points lie within an
arbitrarily small geodesic distance of a.
We shall state a theorem which affirms the existence of a basic deformation
0(t). This deformation will be defined for points on a neighborhood N0 of a
and for a time interval 0 ^ K 1. It will be continuous in that under the def¬
ormation each point p of No will be replaced at the time t by a point q(p, t)
which coincides with p when t — 0, and varies continuously on R with p on N0
and t on its interval. The theorem is as follows.
Theorem 3.1. There exists a deformation B(t) defined and continuous for points
sufficiently near a and for t on the interval 0 ^ t < 1. The deformation 0(t) leaves
points of a invariant and deforms a sufficiently small neighborhood N of a into a
neighborhood Nt, the distance of whose points from a approaches zero uniformly as t
approaches 1. It deforms points below c through points below c .
This theorem is true if / is of class (72 on R, and satisfies certain other general
requirements wrhich do not exclude the possibility of infinitely many distinct
critical sets. In this place we shall give its proof for two general cases. In one
case / will be assumed non-degenerate. In the other case / w ill be assumed
analytic, but not constant. The next two sections will be occupied with this
proof. In the remainder of this section we give certain consequences of the
theorem .
Let N* be a fixed neighborhood of a wffiose closure is interior t o the domain on
which the deformation 0(t) is defined. We state the following corollary of the
theorem.
Corollary 3.1. Corresponding to any neighborhood X of a on Ar*, there exists a
neighborhood M(X) of a so small that M(X) is deformed under 6(t) only on X.
Each k-cycle on M (X) ( below c) will then be homologous on X {below c) to a cycle
{below c) on an arbitrarily small neighborhood N of a. If Zk ~ 0 on N* (below r),
and Zk is sufficiently near a, then Zu ~ 0 on N ( below c).
In this corollary the phrase (below c ) is to be omitted throughout, or read
throughout at pleasure.
152
THE CRITICAL SETS OF FUNCTIONS
[VI]
An ordered pair of neighborhoods VW of <x will be termed admissible if they satisfy
the conditions
V C M(N*), W <Z M(V)
where M( X) is the neighborhood of Corollary 3.1.
We shall have occasion to use the phrase “corresponding to any admissible
pair of neighborhoods VW” many times. For the sake of brevity we shall
replace this phrase by the expression corr VW. With this understood we now
define two basic types of cycles neighboring a. We shall refer to these cycles as
belonging to a.
By a spannable k-cycle corr VW, we shall mean a k- cycle on W, below cf ^ 0
on Wj but oo o on V below c.
By a critical k-cycle corr VW, we shall mean a fc-cycle on W, oc on V to a k-
cycle on V below c.
Maximal sets of spannable or critical cycles corr VW are of importance in that
they depend only on the neighborhood of a, and that we shall subsequently be
able to determine the type numbers ra* of <r with their aid.
The following theorem is an easy consequence of the preceding theorem and
corollary.
Theorem 3.2. Corresponding respectively to any two choices VW and V'W' of
admissible pairs of neighborhoods there exist common maximal sets of spannable or
critical k-cycles on any arbitrarily small neighborhood of a.
It appears from this theorem that the total number, say yk, of cycles in
maximal sets of spannable (k — l)-cycles and critical fc-cycles is independent of
the choice of admissible neighborhoods VW. It will turn out that yk is finite
and that
mk = yk.
The neighborhood functions of the next section are of aid in establishing Theorem
3.1 and determining yk.
Neighborhood functions
4. Let ip be a function of class C2 of the point (x) on R neighboring a point p.
Suppose p is an ordinary point of both / and ip. The gradient of ip is the vector
whose local covariant components are <pif where is the partial derivative of
with respect to x\ The contravariant components of this gradient are then
gi7iPj. See Eisenhart [1]. A regular curve y orthogonal at each of its points p
to the manifold ip = const, through p will be called a ^-trajectory. We are re¬
stricting ourselves here to ordinary points of ip. The differential equations of
the ^trajectories will be given the form
dt gii<fi<Pj
(4.1)
(hj = 1 , * • , m).
[4]
NEIGHBORHOOD FUNCTIONS
153
The denominator of the middle teim is an invariant which is not zero at ordinary
points of p. Along the ^-trajectories we have
(4.2)
dip dx* _
It “ Tt h
so that we can suppose t = <p along these trajectories.
The /-trajectories are similarly defined and represented.
We shall now define the (^-trajectories. Suppose that the gradients of p
and f Sit p are not parallel. By the (<?/)■ -vector at the point p we mean a vector
which lies in the 2-plane of the gradients of p and /, which is orthogonal to the
gradient of /, and which has a magnitude to be prescribed in (4.5). The contra-
variant components X* of this (<p/)-vector will be proportional to
(4.3) gi](ipj + ofj) (i, j = 1, ■ • • , m)
where a is to be determined so that X1 is orthogonal to the gradient of /. This
gives the condition
(4.4) 9x,{fi<Pi + o/t//) = 0,
from which we see that a particular choice of X1 is
(4.5) X{ = g'ig^ifhfkpj - (Phfkfj) (h, k, i, j = 1, • • • , m).
We prescribe the magnitude of X’ by taking it as this vector.
We shall make use of the invariant
Xv» = gijghk[fhfk(prpj - VhfkVxf,] = A(x)
and shall prove the following lemma.
Lemma 4.1. At ordinary points of f arid p at which the gradients of f and p are
not parallel , A(x) ^ 0.
Suppose A(x) were null. Then from (4.3) and the condition XV» = 0 we
have
gi}(<PiPi + — 0 (i, j = 1, * * * , m)f
and combining this condition with (4.4) multiplied by a we find that
(4.6) gtJ[<Pipj + 2 vpxf) + <r2ftf)] — g"\pi + <rfi] [pj + q/,] = 0.
But gij gives the coefficients of a positive definite quadratic form so that (4.6)
holds only if
Pi + <rfi = 0 (i = 1, • • * , m),
contrary to the hypothesis that the gradients of / and p are not parallel. The
lemma is thereby proved.
We note the converse, that A(x) =0 if the gradients of / and p are parallel.
154
THE CRITICAL SETS OF FUNCTIONS
[VI]
The (^-trajectories will now he defined by the equations
(4.7)
dxl
(it
\'(t)
A(x)
X\x)
We see that along these trajectories
d<p XVi
(it A ( x )
(i = 1, • • • , m).
We can accordingly take t = along these trajectories. We also note that
df ^ Xfi s o
eft A (t)
so that/ is constant along (sc/) -trajectories.
A neighborhood function <p(x) belonging to the critical set a off on which / = c,
is now defined as a function with the following properties:
(a) . It is of class C2 neighboring <r.
(b) . It takes on a proper relative minimum zero on a .
(c) . At points near a but not on a, it is ordinary.
(d) . At points near a but not on a at which / = r, the gradients of / and <p
are not parallel.
If (p is a neighborhood function, the locus ip — e is without singularity for e posi¬
tive and sufficiently small. The same is true of the intersection of ip = e and
/ = c, as follows from (d).
We shall exhibit neighborhood functions ip in certain important cases begin¬
ning with the analytic case. WTe state the following theorem.
Theorem 4.1. In the analytic case the invariant function
f = r’fifi (i,j =
is a neighborhood Junction corresponding to any critical set a off.
That ip satisfies the conditions (a) and (b) upon a neighborhood function is
at once evident. We shall finish by proving the following lemma.
Lemma 4.2. Iff is analytic , any analytic function <p which takes on a proper
relative minimum zero on a is an admissible neighborhood function <p.
The function <p of the lemma satisfies (a) and (b). It must then satisfy (c).
For a is a set of critical points of <p, and if <p were not ordinary near a the critical
set a would be a subset of a larger critical set connected to a. But on all con¬
nected critical loci an analytic function is constant. Thus ip would be zero at
some points near a but not on <r, contrary to the nature of a proper minimum.
Thus (c) holds.
Now (d) could fail only at points not on a at which
(4.8) A(x) = 0, / = c.
[4]
NEIGHBORHOOD FUNCTIONS
155
But (4.8) is satisfied on a. Suppose it were satisfied on a larger analytic locus 7
connected with <r. Let h be any regular curve along which (4.8) is satisfied.
On h,f — c so that
(4.9) /.— = 0 (i = 1, •••,«).
I say that on h,
(4.10) *.f = 0 (t = l, •••,«).
This is certainly true on <r, since <pi — 0 on a. At points not on a at which
A(x) = 0 the gradients of / and <p are parallel by virtue of Lemma 4.1, so that
(4.10) follows from (4.9). Thus <p is constant on h and hence on 7. It must
then be zero on 7. From (b) we see that 7 = a. Thus (d) holds.
The proof of the lemma is now complete and the theorem follows directly.
In the non-analytic case a neighborhood function always exists corresponding
to a non-degenerate critical point, as the following theorem states.
Theorem 4.2. ff in terms of a local coordinate system (x), (x) = (0) is a non-
degenerate critical point off , the function
^ = xixi (i =* 1, • • • , m)
is a corresponding neighborhood function .
The function <p clearly satisfies all the requirements upon a neighborhood
function except possibly the one involving gradients. But the relevant relations
of the gradients of / and <p will be unaltered if we use an orthogonal transforma¬
tion of the variables (x) to bring/ to the form
(4.11) f^c = aJ^. + v (fc = l, •••,«)
where an is a constant not zero, and 77 = o(p2), that is, rj vanishes to at least the
second order with respect to the distance p to the origin in the space (x) .
At ordinary points of / and <p a condition that the gradients of / and <p be not
parallel is the following:
(4.12) (<Pifk — <pkfi) G Pifk ~ <Pkfi) — 2(fkfk<Pi<Pi — <Pkfk<Pifi) 7* 0.
We have merely to show that (4.12) holds when/ = c, and (x) 5^ (0) neighboring
(x) = (0). But the right parenthesis in (4.12) is seen to be of the form
(4.13) 8 [a\xkxkxixi — akxkxkaixixi ] + o(p4).
But on/ =* c, upon using (4.11), we see that the expression (4.13) takes the form
(4.14) %a\xkxkxixi + o(pA).
THE CRITICAL SETS OF FUNCTIONS
156
[VI]
The expression (4.14) however does not vanish for (x) sufficiently near the origin
and not (0). Thus <p satisfies condition (d) on a neighborhood function.
The theorem is accordingly proved.
The following theorem will enable us to give a particularly elegant determina¬
tion of the set of non-degenerate critical points equivalent to an isolated critical
point in the analytic case.
Theorem 4.3. If f is analytic and (x) = (0) is an isolated critical pointy the
function <p = is an admissible neighborhood function.
This follows at once from Lemma 4.2.
The determination of spannable and critical cycles
5. We continue with the critical set a. We suppose that is a neighborhood
function corresponding to a . Neighboring a we shall prove the existence of a
basic set of trajectories termed radial trajectories. They lead away from cr
somewhat after the fashion of rays emanating from a point. The first theorem
is the following.
Theorem 5.1. If <p is a neighborhood function for a, then on the domain
H: 0 < <p ^ r,
where r is a sufficiently small positive constant , there exists a i(radiaVJ field of
trajectories , one through each point of H, satisfying differential equations of the form
~ = B\x) (B'Bi * 0),
where the functions Bl(x) are of class Cl on II. These trajectories reduce to (<pf)~
trajectories on f — c. On them t may be taken equal to <p.
The ^trajectories themselves would do except for the fact that they do not in
general reduce to (<p/)-trajectories on f — c. We shall alter the ^trajectories
neighboring / = c so that they will suffice. For the remainder of this proof wre
shall suppose c = 0.
The (/^-trajectories f emanating from / = 0 on H in general form a field F
only for a short, distance from / = 0, depending upon how near <p is to 0 on the
trajectory f in question. (Recall that <p is constant on each trajectory f.) We
shall be precise and say that we can determine a positive function h(a) of class
C 1 for 0 < a g r, such that the field F persists on a trajectory f on which <p = a
where / changes from —h(a) to h(a). We can in fact define h(a) successively
on the intervals
r ^ a >
r
2’
r
2
^ a >
r
V
r -> \ r
— a > — .
4 - 8’
)
and so define h(a ) for r ^ a > 0.
[ 5 ] DETERMINATION OF SPANNABLE AND CRITICAL CYCLES
157
We now let M{u) be a function of u of class Cl, identically one for u 2 > 1, and
zero for u zero, otherwise positive. Our radial trajectories will be defined as
^trajectories except for the points on trajectories £ where/ changes from ~h(<p)
to hQp),. At these exceptional points the differential equations of the radial
trajectories shall have the form
('•') li - x‘<*> + M [i§)j - x‘w> • • • .
where X% and Yl are the functions appearing in (4.7) and (4.1) respectively.
On / = 0 the radial trajectories reduce to the (^-trajectories (4.7). For
/ — ± h{tp) they take the form (4.1). Moreover on them
^ = <PiX'[\ - M] + viY'M = 1 — M + M = 1 (i « 1, ■ • • , m).
(II
This shows that we can take t = <p on the radial trajectories.
The theorem follows at once.
By a radial deformation we shall hereby mean any continuous deformation
•neighboring a critical set cr in which each point moves, if at all, on a radial
trajectory, and two points for which p is initially the same are deformed so that
at the same time the resulting values of <p are the same. With the aid of suitable
radial deformations we can establish the following statements.
(1) . For any two positive constants c and r\ less than r, the domain <p = e
below c is homeomorphic with the domain <p = rj below c.
(2) . Ife < rj, the domain 0 < ^ rj below c can be radially deformed onto the
domain 0 < ^ e below c , leaving the latter domain fixed, and never increasing <p.
(3) . For any closed point set to on the domain 0 < <p S v below c, there exists a
radial deformation on the same domain that leaves the domain <p = -r? below c
fixed, and deforms the point set to onto the latter domain.
We note that these radial deformations deform points below c through points
below c.
We can satisfy Theorem 3.1 by a particular radial deformation defined as
follows.
The radial deformation R(t). Under R(t) the time t varies on the interval
O^Kl. A point on a radial trajectory at which
<P = r — Or (0 g 6 < 1),
shall remain fixed until t = 0, and shall thereafter be replaced by the point on
the same radial trajectory at which
<p = r — tr.
The deformation R(t) thereby defined clearly satisfies the conditions of Theorem
3.1.
The following theorem is also established with the aid of radial deformations.
158
THE CRITICAL SETS OF FUNCTIONS
[VI]
Theorem 5.2. Corresponding to admissible neighborhoods VW of a let e be a
positive constant so small that the domain p S e is on W.
A maximal set of spannable k-cycles corr VW can then be taken as a maximal
set of k-cycles on <p = e below c,, independent on this domain , but bounding on <p ^ e.
A maximal set of critical k-cycles corr VW can be taken as a maximal set of k-
cycles on <p :g e, independent on this domain of cycles on <p = e below c.
The number of cycles in the above sets will be independent of the constant c chosen
as above.
The reader has doubtless observed that the above manifolds <p = e are without
singularity, as are their intersections with / — c.
Classification of cycles
0, Having analysed two basic sets of cycles neighboring the critical set a we
are now in a position to determine the change in cycles with respect to bounding
as one passes from the domain / < a to the domain / < 6. We are supposing
that / = c on a, that a < c < b, that a and b are not critical values off, and that
c is the only critical value of / between a and b. We also suppose that a is a
complete critical set, that is, the set of all critical points at which/ = c.
We admit the possibility that c is either the absolute minimum or maximum
of /. In the former case the domain / < a is vacuous. This case is not excluded
in the following. The reader will observe that in this case certain of the chains
which appear in the following proofs are null, a case again not excluded. As a
convention we understand that a null cycle bounds.
A spannable (A: — l)-cycle k _ i corr VW will be called linkable if bounding
below r. If 4-i is linkable there exists a chain X* below c such that
(0.1) xl — > Zat _ j (below c).
By virtue of the definition of a spannable (k — l)-cycle there also exists a chain
Xl on W, such that
(6.2) x£ Ik- j (on W).
We set
(6.3) a; + \"k = X,,
and term X/; a k- cycle linking h-\, corr VW. More generally we shall term a
A>cycle linking , corr VW, if some subdivision links a spannable (k — l)-cycle
/*_i in the preceding sense. For the sake of simplicity we shall suppose that a
linking /r-cycle corr VW is always given with a division into cells by virtue of
which it links a spannable (k — l)-cycle corr VW. We shall say that X/; belongs
to any critical set to which lk-\ belongs.
We shall now establish three lemmas on linking Avcycles. We begin with the
following.
[6]
CLASSIFICATION OF CYCLES
159
Lemma 6.1. Let (l)k- 1 be a set of linkable ( k — 1 )-cycles corr VW, and let
(A)a be a set of k-cycles linking the respective (k — \)-cycles of the set (l)k-i corr VW.
A necessary and sufficient condition that (l)k~ i be a maximal set of linkable (k — 1)-
cycles corr VW is that (A )k be a maximal set of linking k-cycles corr VW.
We shall first prove the condition sufficient. We assume therefore that
(A)fc is maximal and seek to prove (l)K- \ maximal.
We shall first show that if uk- 1 is any proper sum of cycles of (i)/c-i,
(0.4) Wjfc_! oo 0 (on V below c).
To that end let A* be the sum of the fc-cycles of (A )k which link the respective
cycles of (l)k~ i in the sum uk~ j. Since {\)k is a maximal set of linking fc-cycles
there must exist a spannable (k — l)-cycle Vk-\ linked by A k corr VW. By
virtue of the definition of a spannable (k — l)-cycle corr VW we have
Vk-i 0 (on V below c).
To establish (6.4) it will be sufficient then to show that
(6.5) Uk-i ~ vk~ i (on V below c).
By virtue of the way A k is given as a sum of cycles of ( A) a we have
A, = a; + A l
where \'k and A^ are chains such that
A* — ► Uk —i (on W)
and
A* Uk- 1 (below r).
By virtue of the fact that \k links Vk- 1 we have
A k = z'k + z'k
where z'k and zk are chains such that
z'k — ► vk- 1 (on W)}
z'k — » vk-i (below c).
From our two representations of A/t we see that
(6.6) A k + z'k= A 1 + z"k (mod 2).
But since the right member of (6.6) is a chain below c, the left member of (6.6)
must reduce mod 2, to a chain below c. Moreover
K + z'k “ > 1 + (on W below c),
from which (6.5) and (6.4) follow.
160 THE CRITICAL SETS OF FUNCTIONS [ VI ]
Thus {l)k~ i is a subset of a maximal set of linkable (fc — l)-cycles. It remains
to prove that (l)k-i is a maximal set of linkable (k — l)-cycles corr VW.
To that end suppose (l)k-i contained fewer cycles than a maximal set of
linkable (k — I)-cycles. There would then exist a set (w)*-i of linkable ( k — 1)-
cycles which with the cycles of the set (Z)*_ i would form a maximal set of
linkable (k — l)-cycles. Let (y)k be a set of A;-cycles linking the respective
members of the set (u)*_i corr VW. Any proper sum of cycles of the sets (X)*
and (fx)k will be a linking A>cycle contrary to the assumption that (\)k is maximal.
Hence (l)k- i cannot contain fewer cycles than a maximal set of linkable ( k — 1)-
cycles, and must therefore be a maximal set of linkable ( k — l)-cycles corr VW.
To prove the condition necessary we assume that (Z)*_ i is a maximal set.
If (X)* were not a maximal set, there would exist a larger set of linking k- cycles
which would be a maximal set corr VW. By virtue of the sufficiency of the
condition already established, there would then exist a maximal set of linkable
(k — l)-cydes corr VW which would be a larger set than (/)*_ i contrary to the
hypothesis that (l)k- i is a maximal set.
The condition of the lemma is accordingly necessary, and the lemma is proved.
Our second lemma on linking cycles is the following.
Lemma 6.2. If Lis the domain below c and (X)* a maximal set of k-cycles, linking
corr VW, any k-cycle which is linking corr VW , is homologous on N*+L to a
combination of k-cycles of (X)*, critical k-cycles corr VW , and k-cycles below c.
Let (/)*_ i be the set of (fc — l)-cycles linked, corr VW, respectively by the
cycles of (X)*. Let zk be an arbitrary k- cycle linking a (k — l)-cycle j, corr
VW. By virtue of the preceding lemma we have
(6.7) Uk~i ~ lk- 1 (on V below c )
where 4-i is a proper sum of cycles of the set (l)k-i- Let X* be the sum of the
A>cyeles linking the respective ( k — l)-cycles of the sum lk~\. Now \k can be
represented as in (6.3). Similarly zk can be represented in the form
(6.8) z'k + zk - zu
where z[ is on W and zk is below c , and where
Zk — > Uk~ l, zk Uk-1.
Upon using (6.3) and (6.8) we see that
- X* = (zk - K) + (zl - xl).
Let Wk be the chain on V below c bounded by uk~ i and lk~ i, by virtue of (6 7).
We see that in the congruence
(6.9) (z'k - X* + wk) + (zl - X* - wk) ss zk - K
the first parenthesis is a A;-cycle on V, and the second a fc-cycle below c
[6]
CLASSIFICATION OF CYCLES
161
But any k- cycle on V can be deformed on N* under the deformation S(t) of
Theorem 3.1 into a fc-cycle on W> and hence is homologous on N* to a combination
of critical A;-cycles corr VW and cycles below c. From (6.9) we then conclude
that Zk — A* is homologous on N* + L to a linear combination of critical cycles
corr VW and cycles below c. The lemma is thereby proved.
We shall prove the following lemma.
Lemma 6.3. No k-cycle A* which is a linking k-cycle corr VW is homologous on
TV* + L to a combination of critical cycles corr VW and cycles below c.
Suppose that we had an homology
(6.10) A* + mck + wk~ 0 (on AT* + L)
where m — 0 or 1, cjt is a critical Ar-cycle corr VW, and wk is a cycle on L. Let
wk+ 1 be a chain on N* + L bounded by the left member of (6.10). We can write
wk+ 1 = w'k+1 + w^ , ,
where wk+ x is a chain on TV* and w'kj, x a chain on L, provided, as we suppose is
the case, wk +J is sufficiently finely divided. Thus
wk+ 1 + w"k+ j -4 \k + mck + wk.
Suppose that
w'k+i w'k, wk+1 wk .
Upon using (6.3), and the preceding bounding relations we see that
(6.11) wk + w"k = A^ + A l + mck + wk .
From (6.11) it appears that the chain
Afc + rnck + w'k (on TV*),
reduced mod 2, lies on L, since the remaining chains in (6.11) lie on L, But we
also see that
A* + mck + wk — * lk~ i
where i is the (k — l)-cycle linked by A* corr VW.
We arrive at the conclusion that lk~i lies on W, below c , and bounds a chain
uk on N* below c. Upon applying the deformation 6{t) of Theorem 3.1 to uky
we see that uk will be deformed below c onto V, while lk~ i will not be deformed
off from V. Hence
lk~i ~ 0 (on V below c).
But this is contrary to the fact that lk~i is linked by A* corr VW. Thus (6.10)
cannot hold, and the lemma is proved.
We now define a deformation A (t) related to the deformation 0(t) of Theorem
3.1.
162
THE CRITICAL SETS OF FUNCTIONS
[VI]
The deformation A(t), 0 ^ t < 1. We extend the definition of 6{t) so that the
resulting deformation A (t) is continuous over / < b and remains identical with
6(t) over the neighborhood N* of §3. To that end let e be a positive constant
so small that the set of points not on N* but at a distance at most e from N* are
within the domain of definition of 6(1). Under A(t) each point p at a distance
(1 — X)e from N *, where 0 g A < 1, shall be deformed as in 6(t) until t = X, and
held fast thereafter. Points at a distance e or more from N* shall be held fast
under A(t).
Our fourth lemma on linking cycles is the following.
Lemma 6.4. Let N he an arbitrarily small neighborhood of <r, and L the domain
below c,. Under Aft) any cycle zk which is linking corr VW can be deformed on
V + L into a cycle A* again linking corr VW, and on the domain N + L.
The cycle zk lies on W + L. From the nature of A (t) it is clear that zk can be
deformed on V + L into a cycle \k on N + L. Suppose zk links a cycle uk- 1 corr
VW. Under A(t), uk-\ will be deformed on V into a cycle r*_i. We must have
vk-i ^ 0 (on V below c),
for otherwise
Ujt_i ~ 0 (on V below c),
contrary to the nature of uk->. Hence vk-i is a spannable (k — 1) -cycle corr
VW. Returning to the deformation A(0 we see that \k links vk~\, and the lemma
is proved.
We shall make use of the trajectories
(6.12) ^ = - 9"f*i (i, j = 1, ■■■ ,m)
orthogonal to the manifolds / constant. We make the convention that there is a
trajectory coincident with each critical point at all times t.
The deformation D . Under the deformation D each point on f < b which is
at a point p when t = 0, shall be replaced at each time t for which 0 g t g 1 by
the point t on the trajectory issuing from p. Under D a point which is not a
critical point is so deformed that /is continually decreased. Critical points are
held fast under D. With the aid of D we shall establish a deformation lemma
in the large.
Deformation Lemma. Let N be an arbitrary neighborhood of the critical set a ,
and let L be the set of points below c. A sufficient number of iterations of the deforma¬
tion D will provide a deformation A of the domain f < b on itself onto the domain
N + L.
If a cycle zk lies on a domain No + L for which No is a sufficiently small neigh¬
borhood of (Ty and if zk 0 onf < b (below c), then zk ~ 0 on N + L ( below c).
[6]
CLASSIFICATION OF CYCLES
163
By virtue of the continuity of the deformation D there will exist a neighbor¬
hood TV' of a so small that D will deform A' only on TV.
Let a and P respectively denote the domains / < a and f < b, The domain
P — a — N’ has a positive distance from a, and hence each point p on this
domain will be carried by D into a point at which / is at least d less than at p,
where d is a positive constant independent of p.
Moreover a sufficiently large number of iterations of D will define a deforma¬
tion, say Dr, which will carry p into a point set on the domain
f < c + d/2 .
From the choice of d we see then that Z)r+l will carry all points of p whose rth
images are on p — a — TV' into points on the domain
f < (c + d/2) - d = c - d/2,
while points whose rth images are on TV' will be deformed onto A under Dr+l.
The deformation A = /> H accordingly deforms the domain / < b on itself
onto TV + L.
To establish the final statement of the lemma let Ar0 be a neighborhood of cr
which is so small that No is deformed under A only on TV. Suppose the cycle zk
of the lemma bounds a chain zk+i on f < b. The deformation A will carry zk+\
into a chain on N + L, deforming zk on N + L. Hence if zk ~ 0 on / < b
(below r), it follows that zk ~ 0 on TV + L (below r).
The proof of the lemma is now complete.
Before coming to the principal theorem we define a new set of cycles. A
A;-cycle below c, independent below c of the spannable A>cycles corr VW , is
termed an invariant k-cycle corr VW. Future theorems justify this term.
From the definition of an invariant k- cycle corr VW it follows that a Ar-cycle
below c is dependent on / < b upon a linear combination of invariant A*-cycles.
From the definition of critical A>cycles corr VW it follows that any fc-cycle on
W is dependent on V upon a linear combination of critical fc-cycles corr VW and
k- cycles below c.
We come to a basic theorem (Morse [11, 12]).
Theorem 6.1. A maximal set of k-cycles on f <6, independent on f < b , is
afforded by maximal sets of critical , linking, and invariant k-cycles corresponding
to an admissible pair of neighborhoods VW of the critical set a.
We shall prove the theorem by proving statements (a) and (b). Statement
(a) follows.
(a). Any k-cycle zk on f < b is homologous on f < b to a linear combination of
the k-cycles of the maximal sets corr VW of the theorem.
By virtue of the Deformation Lemma wre lose no generality if we suppose zk
lies on W + L, where L is the domain / < c. If sufficiently finely subdivided,
zk can then be represented in the form
(6.13) zk = zk + zk
THE CRITICAL SETS OF FUNCTIONS
164
[VI]
where zk is a chain on W and z"k a chain on L. Suppose zk~\ is the common
boundary of z'k and zk so that
(6.14) zk — > zk~i, zk — > zk~ i.
We admit the possibility that any one of the chains in (6.14) may be null.
The cycle is necessarily on W below c. It bounds on W and below c.
It accordingly satisfies an homology
(6.15) Zk -i ~ r4-i (on V below c)
where 4_-i is a linkable (k — l)-cycle corr VW, and r — 1 or 0.
Let Xa. be a A>cycle linking 4 -i on W + L. By virtue of (6.15) there exists a
chain wk on V below c. such that
U\ — > Zk-i — 7*4-1.
Upon using (6.3) and (6.13) wre obtain the congruence
zk — r\k = (zk — r\k + wk) + ( zk — r\k + wk).
The first parenthesis contains a &-cycle on V and the second a Zr-cycle below c.
But /c-cycles below c are homologous on f < b to zero or to an invariant Avcycle
corr VW, while A;-cycles on V are homologous on N*, and hence on / <6, to
a linear combination of critical ft-cycles corr VW and ^-cycles belovr c. Thus
zk is homologous on / < b to a linear combination of cycles as stated in the
theorem.
(b). The cycles of the maximal sets of critical , linking, and invariant k-cy cles
corr VW are independent on f < h.
Suppose that there existed an homology of the form
(6.16) m\k + nck + rik ~ 0 ( m , n, r = 1 or 0)
where \k, ck, and ik are respectively linking, critical, and invariant fc-cycles corr
VW. We shall prove successively that m, n, and r are zero.
Proof that m = 0. Suppose m ^ 0. By virtue of the Deformation Lemma
the homology (6.16) implies a similar homology on V + L, provided the cycles
in (6.16) lie on N0 + L, where No is a sufficiently small neighborhood of <r.
But we have seen that X* and ck can be respectively deformed under A(/) into
cycles x; and c[ on N0 + L, where X^ is a linking cycle corr VW and ck is a critical
cycle corr VW. We thus have an homology
m\'k + nck + rik ~ 0 (on / < 6).
By virtue of the Deformation Lemma this implies a similar homology on V + L,
contrary to the nature of the linking cycle X^ as described in Lemma 6.3.
Proof that n = 0. We suppose that m = 0 and n = 1 in (6.16). We then
write (6.16) in the form
(6.17)
2*+i + z*+i — ► nck + rik
[7]
THE TYPE NUMBERS OF A CRITICAL SET
165
where z*+1 is a chain on W and z*+1 a chain on L. Let z[ and z* be respectively
the boundaries of zk+l and zk±x . From (6.17) we see that
(6.18) nek + rit + zk + = 0.
Now zk ~ 0 on Wy and we see from (6.18) that
nek ~ rik + zk (on W),
where the right member is on L. This is contrary to the nature of a critical
k- cycle unless n — 0. Hence n = 0.
Proof that r = 0. Returning to (6.18) with n = 0, and noting that zk ~0on
Lf we have
(6.19) rik~z'k (on L).
If z[ ^ 0 on Ly r must be zero in (6.19), because invariant A>cycles do not bound
belowr c. If zk oo o on Ly z'k is a spannable fc-cyele corr VW, and we again infer
that r = 0, since invariant ^-cycles are independent below' c of spannable k -
cycles corr VW.
Thus in (6.17), m = n = r = 0, and the proof of (b) is complete. The theorem
follows directly.
The type numbers of a critical set
7. In §2 we associated a set G of mk ideal non-degenerate critical points of
index k (k = 0, 1 , * • • , to) with each complete critical set cr, terming this asso¬
ciated set “equivalent” to a, and terming mk the A:th type number of the set a.
This number rnk wras there defined as the sum
(7.1) ?nk = m\ + m~k .
Recall that ?nk is the number of new' A;-cycles and nil the number of newdy-
bounding ( k — l)-cycles in maximal sets of such cycles associated with the
critical value c.
By a newly-bounding k-cycle corr VW wre mean a spannable &-cycle corr VW
which is not homologous to zero below c. It follow's from this definition and
from the definition of invariant (k — l)-cycles corr VW, that a maximal set of
(i k — l)-cycles independent below c, consists of maximal sets of invariant and
newly-bounding ( k — l)-cycles corr VW. Of these cycles the invariant ( k — 1)-
cycles corr VW remain independent on/ < b, according to Theorem 6.1. Hence
to* equals the number of newly-bounding (fc — l)-cycles corr VW in a maximal
set of such cycles. It also follows from Theorem 6.1 that to£ is the number of
critical and linking fc-cycles in maximal sets of such cycles corr VW. Turning
to the definitions of these cycles in §3 and §6 we obtain the following theorem.
Theorem 7.1. The type number mk of the critical set a equals the number of
critical k-cycles and spannable ( k — 1)-cycles in maximal sets of such cycles cor¬
responding to two arbitrarily small admissible neighborhoods VW of a.
166 THE CRITICAL SETS OF FUNCTIONS l VI ]
This theorem is of basic importance in that it shows that the type number
rrik of a depends only upon the nature of / neighboring c r, unlike mX and m T which
in general depend upon / on a larger domain.
The theorem has been proved for the case of complete critical sets. For the
case of critical sets a in general we make the evaluation of mk given by the
theorem serve as the definition of the type numbers of a. If a complete critical
set <j is the sum of a finite ensemble of connected critical sets, as is true in the
analytic case, we see that the type number rnk of a is the sum of the corresponding
type numbers of the component connected sets.
We shall nowr further determine the type numbers mk in the most important
cases. The following evaluation of mk makes use of configurations defined by /
and neighborhood functions of a. It depends upon Theorems 7.1 and 5.2. In
it e is an arbitrarily small positive constant.
I. If a is a connected critical set possessing a neighborhood function (py the number
mk is the siim of the numbers of cycles in the following two sets:
(a) . A maximal set of (k — \)-cycles on <p — e below cy independent on this
domain , bounding on<p ^ e.
(b) . A maximal set of k-cycles on <p ^ ey independent on <p ^ e of the k-cycles on
^ = e below c.
We term a critical set on which / takes on a proper relative maximum or
minimum, a maximizing or minimizing set respectively. For a maximizing or
minimizing set on which / = Owe note that the functions — / and / are respec¬
tively admissible neighborhood functions, and for such sets I holds with <p = — /
and ^ = / respectively. In particular we note that for a minimizing set, mk is
the kth connectivity of the domain/ g e neighboring a.
Concerning the numbers m0 and mm we have the following theorem.
II. The type number m0 is 1 for each connected minimizing sety and null for all
other connected critical sets. The type number mm is l for each connected maximiz¬
ing set and null for all other connected critical sets.
By virtue of Theorem 7.1, m0 is the number of critical 0-cycles in a maximal
set of such cycles corr VW. If a is connected, any two of its points can be
arcwise connected in any arbitrarily small neighborhood of a so that there is at
most one 0-cycle in a maximal set of critical 0-cycles. If a is not a minimizing
set, there are points arbitrarily neai a at which/ < c, so that corresponding to N,
any point of a sufficiently small neighborhood No of cr can be arcwise connected
on N to a point on N0 below c. Hence m0 is 0 for connected critical sets which
are not minimizing sets; m0 is 1 for each connected minimizing set.
Before turning to mmy let it be assumed that R is an m-circuit, that is, possesses
no sub-complex of cells which is an m-cycle, and that R satisfies the manifold
condition that any chain Cm of m-cells of R which contains a point P of Ry either
contains all cells incident with P or else possesses a boundary (m — l)-cell
incident with P.
We shall show that when mm > 0 the set a must be maximizing.
Since R is an m-circuit, any m-cycle sufficiently near a is bounding near or, so
[7]
THE TYPE NUMBERS OF A CRITICAL SET
167
that there are no critical ra-cycles. The number rnm must then be the number of
spannable (m — l)-cycles in a maximal set of such cycles corr VW.
.Suppose that zm- 1 is such a spannable ( m — l)-cycle corr VW, Without loss
of generality we can suppose that zm-i is composed of cells of a subdivision of R
and bounds a chain zm of such cells on V, because in any case we could use the
Veblen- Alexander deformation to replace zm by a nearby chain of that nature.
On zm~\,f < c , and on zm there are points at which/ ^ c. Let gj be the set of
points on zm at which / takes on its absolute maximum relative to its values on
zm. Each point P of a x will afford a relative or absolute maximum to / on R ,
since zm contains an entire neighborhood of P on R. Hence g\ <Z a. The set
a\ is closed and, since / is constant on a, contains all points of g neighboring any
point P of <ti. The set a i must then be identical with g, since g is connected.
If mm > 0, the set g must then be maximizing.
It remains to prove that rnm = 1 if g is a connected maximizing set.
Let zm be the set of all m-cells of R whose closures contain points of g. If R is
sufficiently finely subdivided, zm will lie on W . Its boundary zm-i will be below
c. I say moreover that zm- j will not bound on V below c. For, by virtue of the
Veblen-Alexander process, zm-i would then bound a chain z'm of cells of a subdivi¬
sion of R below c. We suppose that zm and z'm consist of cells of a common sub¬
division of R. The sum
■ f
Zm I Zm}
reduced mod 2, will then be a non-null, non-singular m-cycle of cells of R covering
at most a neighborhood of o-, contrary to the hypothesis that R is an ra-circuit.
Thus Zm-i does not bound on V below c. Hence zm_i is spannable corr VW and
mm ^ 1.
Finally I say that mm = 1. To prove this let wm _] be a second spannable
(m — l)-cycle corr VW, consisting of cells of R, and bounding a chain wm of
cells of R on V. We suppose moreover that zm and wm belong to a common sub¬
division, say R\ of R. We have
Wm Zm ' Wm — 1 2m— 1*
By virtue of the manifold property of R', as previously assumed, both wm and
zm must contain each m-cell of R whose closure contains a point of cr, and except
for these cells consist of points below c. Hence wm — zm reduces, mod 2, to a
chain below c. Hence
Wm- 1 ~ Zm- 1 ~ 0 (on V below c ).
Thus mm = 1 for a connected maximizing set, and the proof is complete.
The first of the following results was stated by A. B. Brown, but not com¬
pletely proved by him (Brown [1]).
III. Suppose (x) — (0) is an isolated critical point in a coordinate system ( x )
in which f is analytic. If we set <p — xixi , the jth type number m, of ( x ) = (0) is
given by the formula
(7.2) m, = #,_! - 8[ (j > 0)
168
THE CRITICAL SETS OF FUNCTIONS
[VI]
where Rj is the jih connectivity of the domain <p — e below c. In the case of a mini¬
mum Wo = 1, otherwise Wo = 0.
The type numbers are similarly evaluated if f is merely of class C 2 and (x) — (0)
is a non-degenerate critical point of f.
Relations (7.2) follow from I upon determining the relevant critical and
spannable cycles.
First observe that there are no critical ^-cycles when k > 0, since for k > 0
all A>cycles which lie on <p — e bound on <p g e. Turning to spannable (k — 1)-
cycles we observe that all ( k — l)-cycles on (p = e are bounding on v g e when
k — 2, • • • , m. Hence mk — Rk_} when k — 2, • * * , w. To determine Wi we
observe that there are Ro — 1 spannable 0-cycles independent on <p — e below r,
each 0-cycle consisting of a pair of points. Hence wi = R0 — 1. Finally to
determine w0 we use II. We conclude that w0 = 1 in the case of a minimum.
Otherwise w0 = 0.
The second paragraph under III gives a preliminary determination of mk
in case the critical point is non-degenerate. The final result in this case is the
following.
Theorem 7.2. The jth type number of a non-degenerate critical point of index k
equals 8Jk where 8k is the Kronecker delta ( k,j = 0, 1, * • • , w).
We suppose that (x) = (0) is the critical point and that/(0) = 0. If k = 0,
the critical point affords a relative minimum to f and the theorem follows from
III. If k > 0, and j = 0, w0 = 0 according to III and the theorem is again true.
If k > 0 and j > 0, we begin by making a non-singular, linear, homogeneous
transformation T of the local coordinates (x) to local coordinates (y) of such a
nature that / takes the form
/(T) = - y\ - ■ ■ ■ - y\ + yl+1 + • • • + yl + oi(yh ■■■ , ym),
where w is of more than the second order with respect to the distance to the origin
in the space (y). We now regard / as a function F(y) of the variables (y).
Under the transformation T a neighborhood function will be carried into a
neighborhood function, and it follows then from I that the type numbers of
(a:) = (0) as a critical point of f(x) equal those of ( y ) = (0) as a critical point of
F(y).
WTith F we now consider the 1-parameter family of functions
F(y, a) = — y\ — — yl + yl+ 1 + • • • + yl + ^(.y)
where n is a constant on the interval 0 S m ^ 1. For each value of y, F(y , y)
has a non-degenerate critical point of index k at the origin, and the function
<P = 2 Wi (t = 1, • • • , w)
is a corresponding neighborhood function, provided
ViVi ^ r,
[7]
THE TYPE NUMBERS OF A CRITICAL SET
169
where r is a sufficiently small positive constant. Reference to the proof of this
fact in §4 shows that this constant r can be chosen independently of the choice of
P on the interval 1. But we have seen under III that the type number
nij of ( y ) = (0) as a critical point of F(y, p) is given by the formula
rrtj = - <5 { (j > 0)
where Rj is the^’th connectivity of the domain determined by the conditions
<P = r, F(y> p) < 0.
We continue with the following lemma.
Lemma 7.1. The domains are homeomorphic for all values of p on the interval
0 ^ p S 1, and the type numbers of ( y ) = (0) as a critical point of F(y} p) — 0 are
accordingly independent of p.
To prove this lemma observe first that Xfs boundary is the domain
V = r> F(y , p) = 0,
and is without singularity, since tp is a neighborhood function. Let p0 be a
particular value of p on the interval (0, 1), and set
F(y, Mo) = tip).
If pi is a second value of p sufficiently near p0, one can use the (^)-trajectories
of §4 to show that the domains ZMo and are homeomorphic. To that end one
considers a (i/y>) -trajectory 77 through each point of B and takes ^asa param¬
eter on this trajectory. If e is a sufficiently small positive constant, points on
the trajectory 77 for which
— e ^ f g c
will form a field H on <p = r neighboring B^. Moreover one shows readily that
for pi sufficiently near p0 there is one and only one trajectory of H through each
point of B^y and that the point of intersection of 77 with B Ml varies continuously
with its intersection with B M0.
Suppose that pL is taken so near p0 that each trajectory 77 meets B in a point
(y) at which \p equals a value \p , such that
- e < ir, < e.
We nowT establish a homeomorphism between and 2Ml by making each point
( y ) on 2W and 77 for which
(7.3) ^ *(V) ^
correspond to that point ( y ) on 2„0 on the same trajectory at which \p(y) divides
the interval (0, — e) in the same ratio as that in which \p(y) divides the interval
(7.3) .
We make the remaining points of 2Mo correspond to themselves. We have thus
170
THE CRITICAL SETS OF FUNCTIONS
[VI]
established a homeomorphism between ZM0 and SMl for /xa on a sufficiently small
open interval including y0. But the whole segment 0 ^ y g 1 can be covered
by a finite set of such intervals, from which it follows that the domains are
homeomorphic for 0 ^ y ^ 1.
The lemma is thereby proved.
The type numbers of (x) = (0) as a critical point of/ are accordingly the same
as the type numbers of ( y ) = (0) regarded as a critical point of the function
(7.4) Q(y) = - y\ - y\ + y\+ x + • • • + yl (ft > 0).
To determine these type numbers according to III, we have merely to determine
the connectivities of the domain
(7-5) y\ + • • • + yl = e, Q(y) < 0 (k > 0),
where e is a positive constant. We come then to the following lemma.
Lemma 7.2. The connectivities of the domain (7.5) are those of the (k — \)-sphere.
The connectivities of the domain (7.5) are those of the domain
(7.6) 0 < y\ + • * * + yl ^ e, — y\ — • • • — y\ + y\ M + ■ • • + yl < 0,
since every chain pr cycle on the domain (7.6) can be radially deformed on (7.6)
into a chain or cycle on (7.5). But the domain (7.6) can in t urn be deformed on
itself into the configuration
(7.7) ?/£ + 1 + * • • + Vm = 0, 6 < y\ + • • • + yl g e,
as follows. Corresponding to each point ( y ) = (a) on (7 .6) we hold
(y b * , Vk)
fast and deform the point ( y ) in such a manner that the variables
+ 1 I , • • • , I Vm |
decrease to zero at rates respectively equal to their initial values
| && + 1 j y * ) | 0,m | .
As a final simplification we radially deform the configuration (7.7) on itself into
the (k — 1) -sphere
(7.8) yl+i + ■■• + yl = 0, y\ + ■■■ + y\ = e.
The connectivities of the configuration (7.5) are then those of the (k — 1)-
sphere (7.8), and the lemma is proved.
In Lemma 7.1 we have seen that the type numbers of ( x ) = (0) as a critical
point of /equal those of (y) = (0) as a critical point of the form Q(y) of (7.4).
By virtue of III the latter type numbers are given, for / > 0, by the formula
m j — R j — i 5 1 ,
[7]
THE TYPE NUMBERS OF A CRITICAL SET
171
where ft,_ i is the ( j — l)st connectivity of the domain (7.5), or, according to
Lemma 7.2, the (J — l)st connectivity of the (k — l)-sphere. Hence rrij =
where k is the index of the critical point, and Theorem 7.2 is proved.
Theorem 6.1 leads at once to the following theorem. In it the domain f < a
(or f < b) may be vacuous, and in such a case we understand that its con¬
nectivities are null.
Theorem 7.3. Let a and 6, a < b , be any two constants which are not critical
values of /. Let ARk denote the kth connectivity of the domain f < b minus that of
the domain f < a . Let Mk be the sum of the kth type numbers of the critical sets a
on the domain a < f <6. Finally let M\ be the number of cycles in the ensemble
of maximal sets of new k-cycles relative to the different critical values of f between a
and 6, and, Mf be the corresponding sum for newly-bounding (k — 1 ) -cycles. Then
(7.9)
ARk = Mt - (k = 0, 1, • • • , m),
Mk = Mf + M7,
where M 0 = M“+1 = 0.
If we eliminate the integers Mf from the relations (7.9), we obtain the follow¬
ing m + 1 relations (i — 0, 1 , • * • , m) :
(7.10) Mo - Mi + • • • + ( - 1 YMt = A(R0 - Rx + ■ ■ * + (- 1)%) + (-^Mf, 1 .
In particular for i = nt, we have the relation
Mo - M, + • • • + (-1 )mMm = A(ft0 -«!+••-+ (-l)miO.
Theorem 7.3 holds in particular if a is less than the absolute minimum of /
on ft and b is greater than the absolute maximum of / on ft. For this case
(7.10) takes the form
(7.11) Mo~ M,+ ... +(-l)''M. = ft0-ftl+ +(-l)‘ftt+(-l)iM;M
where the connectivities ftt are those of the whole manifold ft. From (7.11) we
then have the following theorem.
Theorem 7.4. Between the connectivities ft, of the Riemannian manifold ft and
the sums Mi of the ith type numbers of critical sets off , the relations (1.1 ) of Theorem
1.1 hold. In particular the validity of Theorem 1.1 is established.
We point out that the two important corollaries of Theorem 1.1 hold with the
present interpretation and evaluation of ill,. These corollaries form the basic
means of establishing the existence of critical points.
We also state the following corollary of Theorems 7.2 and 7.3.
Corollary 7.4. If c is the only critical value between a and b, and is taken on
by just one non-degenerate critical point P of index k , then
Mi = hi
172
THE CRITICAL SETS OF FUNCTIONS
[VI]
and the ith connectivity of the domain f < b minus that of the domain f < a affords a
difference &Ri which is zero except that either
A Rk = 1
or
A/4-i = -1.
We shall say that the critical point P is of increasing type if A/4 — 1, and of
decreasing type if A/4-i = — 1 . We see that P will always be of increasing
type if k = 0. We shall make use of the following remark in subsequent work.
Remark. If k > 0, P will be of increasing type if and only if there is a linkmg
k-cycle associated with P.
This follows from the fact brought out in the proof of III of this section, that
for A; > 0 there are no critical cycles associated with an isolated critical point.
The following theorem is useful in later work.
Theorem 7.5. Let *p(x1 2f * • • , xm) be a function which is analytic neighboring
the origin in the space (x), which vanishes at the origin , and there possesses a non¬
degenerate critical point of index k, 0 < k g m. Let X be a regular analytic
k-dimensional manifold which passes through the origin , and on which <p(x) has a
proper maximum at the origin. Corresponding to any sufficiently small neighbor¬
hood N of the origin there exists a positive constant e so small that the cycle defined
by ^ = —e on 2 is non-bounding on N among points at which <p < 0.
Before giving the proof we remark that the theorem can be shown to be false
if S is not regular.
We begin with the following lemma.
Lemma 7.3. There exists a non-singular analytic transformation of the variables
(x) neighboring (x) = (0) into variables (?/) neighboring (y) = (0), which carries
(x) = (0) into ( y ) — (0), and under which
(7.12) <p(x', y* + y\ + 1 + ... +yl
According to Taylor’s Theorem we can write <p{x) in the form
(7.13) <p(z) = ciijix)!*!’ (i, j — l, ■ ■ ■ , to),
where
r 1 ^2
«w'W = I (1 - u) (uxl, ■ ■ ■ , uxm)du.
See Jordan, Cours d’ Analyse, vol. I, p. 249. We note that a ,-,•(: r) is analytic in
(x) for (x) near (0), that it is symmetric in i and j, and that
a,j(0) =
1 3V(0)
2 dx*dx’ '
THE TYPE NUMBERS OF A CRITICAL SET
173
If an (0) 7^ 0, we make the transformat ion
_ flaws'
| ttnOr) | 1/2
- JT’
U = 1, * * * , m),
(j = 2, • • • , m),
as in the Lagrange transformation of quadratic forms. One then verifies the
fact that
<p =
+ Q(Z'2, ” ’ i -m).
Here Q(z ) is of the same form as <p(x) in (7.13), involving merely the variables
z2) • • • , cTO. If the coefficient of z\ in Q{z) is not zero at ( z ) = (0), we make a
similar transformation of the variables z 2, • • • , and so on until we have
reduced <p(jr) to a form involving squares only of the variables, with coefficients
which are db 1. The transformations involved have all been non-singular and
analytic, neighboring the origin.
If a 1 1(0) =■ 0, at least one of the remaining coefficients ar« will not be zero
since | atJ(0) | 3^ 0. If the preliminary transformation
Xr — Zr ZH,
3 Zr + ,
= Zi (i 9^ r, s)
be made, the resulting coefficients of z\ and z\ will not be zero at. the origin, and
upon taking zT as zY we proceed as before.
Finally after a suitable relettering of the variables, will be reduced to the
form (7.12). The number of minus signs on the right must thereby be exactly k .
For if we transform the ordinary quadratic form which gives the terms of second
order in y? by using merely the linear terms in the preceding transformations,
the quadratic terms in <p will be carried into the form on the right of (7.12).
According to the classical law of inertia for quadratic forms the number k is
invariant under such transformations.
The proof of the lemma is now complete.
We continue the proof of the theorem by establishing the following statement,
(a). If the manifold 2 is represented regularly and analytically neighboring the
origin by a power series in k parameters (w), in the form
(7.14) y , = btJUj + • • * (? = 1, • • ■ , m; j = 1, • • • , fc),
then
(7.15)
| bhJ- | 0
(h7j =!,■••,&)•
Since the representation is regular, the complete matrix
II bit l| (i = 1, • • * , rn;j = 1, * • • , k)
174
THE CRITICAL SETS OF FUNCTIONS
[VI]
must have the rank k. I say in particular that (7.15) must hold. For otherwise
there would exist constants not all zero, such that
bhjCj = 0 (hfj = 1, • • • , k).
Upon setting Uj = tc} in (7.14) and evaluating <p upon the resulting curve y on
2, we would find that at t = 0
^ = 0, ^ 2 (KciY > 0 (h = k + 1, ■ ■ ■ ,171] j = 1, • • • , k),
k
so that <p would have a minimum on y at the origin, contrary to the hypothesis
that (p has a proper maximum on 2 at the origin. Hence (7.15) holds as stated,
and statement (a) is proved.
We now introduce a deformation <5 with the following property:
(b) . If a is a sufficiently small 'positive constant , there exists a continuous deforma¬
tion 8 of the domain
(7.16) y\ + • • • + vl S «2, <p < 0,
on itself onto its subdomain 2a on 2 .
The deformation 8 will be defined as the product of two deformations, p
and y.
The deformation fi. In defining ft we naturally restrict a to values so small that
for yiyi ^ a2 the representation (7.12) holds. Under p each point ( y ) on (7.16)
shall be continuously deformed into a point on the domain
(7.17) 0 < y\ + • • • + y\ g a\ yi+1 + • • • + yl = 0,
by holding yu ••• , yk fast, and letting the variables
| Vk+l | , * ’ * y I Vm |
approach zero at rates equal to their initial absolute values. Under p points at
which v? < 0 will be deformed through such points.
The deformation y. By virtue of statement (a), points (; y ) on 2a will be
determined in a one-to-one continuous manner by their first k coordinates
(2/1, • • • , yk)y provided the constant a in (7.16) is sufficiently small, as we suppose
is the case. As applied to 2a, the deformation p replaces 2a, at each instant of
the deformation, by a one-to-one continuous image of 2a. The deformation d,
as applied to 20, then has a unique inverse P~l which we could apply to (7.17)
to deform (7.17) onto 2„, except that (7.17) might thereby be deformed outside
of (7.16). To avoid this difficulty we first deform (7.17) radially on itself into a
similar domain so near the origin that the resulting points of (7.17) are deformed
under P~l on (7.16) onto 2a. We term the resultant deformation of (7.17) onto
2 a, the deformation y.
The deformation 8 = Py clearly has the properties ascribed to 8 in (b), and (b)
is accordingly proved.
[81
THE COUNT OF EQUIVALENT CRITICAL POINTS
175
We now return to the theorem, and let r be a positive constant so small that
the domain excluding the origin and including the points
(7.18) tp ^ - r (on 2),
neighboring the origin and connected to the origin, is without singularity or
critical point of <p regarded as a function of the point on 2. Let e be any positive
constant less than r. The cycle <p = —e on 2 will be non-bounding on (7.18)
below 0. For if the cycle <p = — e were bounding on (7.18) below 0, a use of the
trajectories on 2 orthogonal to the manifolds <p constant on 2 would show that
the cycle <p — —e on 2 would be bounding on itself, which is impossible.
With r so chosen we choose the constant a as previously, with the additional
restriction that the intersection of 2 and the domain
(7.19) xjiyi ^ a2 (i = 1, • • • , m)
be interior to the domain (7.18). Let N then be any neighborhood of the
origin on the domain (7.19). Let e be any positive constant less than r, and
such that the cycle <p = — e on 2 is on N. I say that this cycle will then be
non-bounding on N below 0.
For if the above cycle bounded on N below 0, it would bound below 0 on the
intersection of (7.19) with 2 by virtue of (b). The above cycle would thus
bound on (7.18) below 0, contrary to the nature of the domain (7.18).
The cycle <p = —e on 2 is accordingly non-bounding on N below 0, and the
theorem is proved.
Justification of the count of equivalent critical points
8. W e are counting a critical set with type numbers m0y mh • * • , rnm as equiva¬
lent to a set G of ideal non-degenerate critical points in which the number of
points of index k equals m* (k = 0, 1, * • • , m). In justification of this count we
affirmed in §2 that these numbers rrik or this set G had four properties 1, II, III,
IV. Of these properties, I, II, and III have already been established. It
remains to confirm property IV.
To establish property IV we first prove a number of lemmas. In the first
lemma we shall be concerned with two ordinary values of/, a and 5, with a < b.
A fc-cycle on/ < a, non-bounding on/ < a, but bounding on/ < 6, will be called
a newly-bounding A;-cycle relative to the change from a to b. A fc-cycle on f <b,
independent on/ < b of ^-cycles on / < a, will be called a new k- cycle relative
to the change from a to b. In the following lemma and its proof it will be
convenient to abbreviate the phrase “the number of fc-cycles in a maximal set
of k- cycles” by the phrase the count of k-cycles.
Lemma 8.1. The sum Mk of the kth type numbers of the critical sets with critical
values between a and b will exceed or equal the count u of new k-cycles plus the
count v of newly-bounding ( k — \)-cycles relative to a change from the domain
f < a to the domain f < b.
176
THE CRITICAL SETS OF FUNCTIONS
[VI]
Let
(ii < a2 < * * * < ar (a i = a; ar =* 6)
be a set of ordinary values of / so chosen as to separate the critical values of /
between a and b. Let
fa 0 = 1, • ■ • , r - 1)
be the count of (k — l)-cycles on / < a*, independent on / < a„ bounding on
/ < Lot h\ be the count of such cycles, dependent on / < a, upon cycles
on/ < a. We have//' ^ /it and
r = y] h\ ^ y] hi (i = 1, • • • , r - 1).
> I
On the other hand let be the count of /^-cycles on / < independent on
/ < «l+1 of cycles on / < a,. Let q{ be the count of such cycles independent on
/ < 6 of cycles on / < at. We have q[ rg qt and
s*-
»
Combining these results we find that
U + r = ^ y <7i — A/*
I *
and the lemma is proved.
The second lemma concerns the function <f> of the statement IV of §2. By
hypothesis,
* = /, (a) = (0).
Lemma 8.2. If a and b are ordinary values of f with a < b , then for (ju) sufficiently
near (0) the domains f < b and 4> < b are homeomorphic under a transformation of
K by virtue of which the subdomains f < a and 4> < a are likewise homeomorphic.
The homeomorphism whose existence is affirmed in the lemma is taken as the
identity except for points neighboring f = a and / = b. Neighboring these
manifolds we utilize the trajectories orthogonal to the manifolds / constant to
complete the homeomorphism in the desired manner.
We note that the lemma is also true if either a or b is outside the interval of
values which / takes on.
Now let a be a critical set of/. Suppose / = c on a. The function/ may take
on the same value c on other critical sets. This possibility makes the proof of
IV of §2 more difficult. We can however meet the difficulty by altering/ or
slightly neighboring the critical sets in accordance with the following lemma.
In this lemma we again use the invariant function
<P = <7 i}fzifzi
= Y!
7. ^
0, j = 1 , ■■■ ,m).
[8]
THE COUNT OF EQUIVALENT CRITICAL POINTS
177
Lemma 8.3. Corresponding to any critical set a of f and arbitrarily small positive
constants e , eu and p, e > eL, there exists a function 'k of class C 2 on R, with the
following properties:
(1) . Except when <p < e neighboring o-, 'k = 0.
(2) . When ip < ei neighboring a, p.
(3) . For (p) sufficiently near (0) the function <f> + 'k has no other critical points
than those of <f>.
Let r be a positive constant so small that among the points connected to a
for which
Og^r,
ip = 0 only on <r. Choose e < r.
Let H(z) be a function of a single variable z, of class C2 for z ^ 0, and such
that
U(z) si (0^^ eO,
II (z ) =; 0 (z ^ e).
Neighboring a the function 'L will be defined by setting
'k = pH (<p) (ip g e).
We then take s () elsewhere on R. One sees that 'fr has the properties (1)
and (2). Moreover property (3) could fail only when
(8.1) ei < ip < e.
But for p = 0 and (p) = (0) we have
<*> + * s /.
Moreover <t> + ^ is of class C2 in (x), (p), and p. On the domain (8.1) the gra¬
dient of/ is not null. Accordingly for p and (p) sufficiently near p = 0 and (p) = (0)
respectively, the gradient of <l> + 'k is not null.
Thus <l> + 'k satisfies (3), and the lemma is proved.
If a is any critical set off we shall now justify our definition of the type numbers of
<t and the set of ideal non-degenerate critical points equivalent to a by establishing
proverty IV of §2.
By virtue of Lemma 8.3 we lose no generality if we suppose that the critical
value c taken on by / on <r is assumed on no other critical sets of /. For the
addition to of functions such as ^ in Lemma 8.3 will not change the position or
type numbers of the critical sets of <£ for (p) sufficiently near (0), but will enable
us to make the critical values, belonging to sets other than a, different from the
critical value c.
Suppose c is separated from the other critical values of / and from d= <*> by
constants a and b. We include thereby the special cases where a is less than the
absolute minimum of /, or b is greater than the absolute maximum of /. Let
178
THE CRITICAL SETS OF FUNCTIONS
[VI]
be the set of critical points of $ which lie in the neighborhood of <r, for sets (p)
near (0), but not (0). Let (p) be taken so near (0) that the critical values of <f>
on lie between a and 6, and so that the homeomorphism of Lemma 8.2 holds.
The type number mk of a as a critical set of/ will equal the number, say Nk , of
newly-bounding ( k — l)-cycles and new A:-cycles in maximal sets of such cycles
taken relative to a change from the domain f < a to the domain f < b. By
virtue of the homeomorphism of Lemma 8.2, the number Nk will be the same if
taken relative to a change from the domain <£ < a to the domain $ < b. But it
follows from Lemma 8.1 that the number Mk of critical points of index k in the
set <rM will exceed or equal Nk. Thus
Mk ^ Nk = re¬
statement IV of §2 is thereby proved.
The following theorem gives further content to the preceding theory.
Theorem 8.1. Let a be a critical set of f which lies in a coordinate system (x) in
which f is analytic . Corresponding to any arbitrarily small neighborhood N of a,
there exists a function <£ of class C 2 on R which with its first and second partial
derivatives approximates f and its first and second partial derivatives arbitrarily
closely over R, and, which is such that
f ss $ (on R — N )
while has at most non-degenerate critical points on N .
In the coordinate system (x) consider the functions
F(x, p) = f(x) + mx* (t = 1, • * • , m)
where (p) is a set of constants not (0). The condition that this function F(x, p)
have at most non-degenerate critical points in the system (x) is that the equations
Li + = 0
have no solution in the system (x) at which the hessian of/ vanishes. That a
choice of the constants (p) can be so made arbitrarily near (p) — (0) follows from
a theorem formulated by Kellogg [1]. Let the constants (p) then be restricted
to such sets of constants.
To define the function <£ of the theorem we make use of the neighborhoods
ip < e and <p < ci of <r as in the proof of Lemma 8.3, supposing that ex < e, and
that e is so small that points on <p < e neighboring a lie on N and in the coordinate
system (x). We then use the function H(z) of the proof of Lemma 8.3 and neigh¬
boring <7 set
$ * / + paWfo) (ip s; e ),
taking <£ as identical with/ elsewhere on R. On the neighborhood (p ^ e\ of a
we have
$ 35 / + PiX*
(t * !>•••> *»).
[9]
NORMALS FROM A POINT TO A MANIFOLD
179
The remaining properties of are verified as in the proof of Lemma 8.3, provided
of course the constants (n) are sufficiently near (0).
Normals from a point to a manifold
9. Suppose that our manifold R lies in a euclidean space of m + 1 dimensions.
The problem of determining the straight lines issuing from a prescribed point 0
and normal to R belongs to the theory of critical points of functions as well as to
differential geometry in the large. Since the distance function is involved it is
also a special problem in the calculus of variations in the large.
Suppose first that O is not on R . Then the distance from 0 to R will be a
function / of the point on R, analytic if R is locally analytic, of class C 2 if R is
represented locally by means of functions of class C2. One sees at once that a
necessary and sufficient condition that a point P on R be a critical point of / is
that P be the foot of a normal from 0.
Let P be a critical point of /. Let P be taken as the origin in the space (xlf
■ • * , xm +i), and the m-plane tangent to R at P as the m-plane xm+i = 0. After
a suitable rotation of the xh • • • , xm axes in t he m-plane + i = 0, R can be
represented as follows neighboring P:
(9.1) 2xm4 , = b,x] + H(xu ■■■ ,xm) (* = 1, • • • , to).
Here b ? is a constant, and the first and second partial derivatives of H vanish at
the origin.
For such of the constants bi as are not zero we set
and call the point Pi on the xm+i axis at which xm+i — r* a center of principal
normal curvature or a focal point of R corresponding to P. If a constant bi = 0,
we say that the corresponding center Pi is at infinity. We define the centers P»
for any axes obtained from our specialized set by a rotation or translation, by
imagining each normal to R rigidly fixed to P, and each center Pi rigidly fixed to
its normal. The positions of these centers of curvature will be independent of
the particular coordinate system into which we have rotated the original system
in (9.1).
We shall prove the following lemma.
Lemma 9.1. The index of a critical point P of f corresponding to a normal OP
equals the number of focal points corresponding to P between 0 and P exclusive .
The point P is a degenerate critical point if and only if 0 is a focal point correspond¬
ing to P.
If we use the coordinate system in (9.1) it appears that the point 0 must lie
on the Xm+i axis, say at a point at which xm+i = a. For simplicity suppose
180
THE CRITICAL SETS OF FUNCTIONS
[VI]
a > 0. Neighboring the origin let / be represented by its value /(.ri, • • ■ , ,r,„)
in terms of the coordinates (xi, ■ ■ ■ , xm). For xm+, given by (9.1) we have
$ = (x,Ti + (jm+x - a)2)172
«[l + r,x, ^ + K(x1} • • • , 2
= (i + \ Jr
+ L Oi, • - • , xm)
where K and L are functions of the same nature as H .
The index of the function / at P is accordingly the number of the coefficients
(9-2)
(1 - bi<i)
O' = 1, • • • , m)
which are negative. One sees that the coefficient (9.2) is negative if and only
if bi 7* 0 and 0 < rt < a, that is, if the corresponding focal point lies between
0 and P. Moreover the critical point will be degenerate if and only if one of the
coefficients (9.2) is zero, that is, if some = a and the corresponding focal point
lies at 0.
The lemma is thereby established.
We accordingly have the following theorem (Morse [5]).
Theorem 9.1. Suppose 0 is not n focal point of R. Of the straight line segments
from 0 to R which are cut normally by R at their ends P on P, let Mk be the number
upon which there are k focal points of P between 0 and P. Then between the
numbers Mk and connectivities Ri of R the relations (1.1) of §1 hold.
If 0 is on R, one must count the point 0 as a special normal segment upon which
there are no focal points of R.
It is thereby understood that in counting the number of focal points of P on
OP, a focal point must be counted the number of times that the corresponding
coefficient bi appears in (9.1). Moreover two normal segments OP\ and OP*
are to be counted as distinct if Pi ^ P2 even if OPi and OP2 lie on the same
straight line.
We have the following corollary of the theorem.
Corollary. If 0 is not a focal point of R , there will be at least Ri normals OP
from. 0 to R with i focal points of P on OP , and all told at least
Ro + Pi + * * * + Pm
normals from 0 to R.
As an example suppose R is an orientable surface of genus p. Then
Ro — 1, Pi — 2p, P2 — 1,
so that in the non-degenerate case there will be at least one normal OP from 0
to P with no focal point of P on OP, at least 2 p normals with 1 such focal point
SYMMETRIC SQUARES OF MANIFOLDS
181
[10]
thereon, and at least 1 normal with 2 such focal points thereon. From the
relation
M o — M ) ■+• M 2 — J?0 — R i 4~ R2 — 2 — 2p,
one sees moreover that the number of normals OP is always even.
If 0 is a focal point of R , degenerate critical sets enter, and these must be
counted according to their type numbers.
Symmetric squares of manifolds
10. Let R be a regular analytic m-manifold in a euclidean (m + l)-space £
of coordinates (x). We shall make a study of the critical chords of R , that is,
chords which are normal to R at both of their ends. The function involved is
the length of a variable chord whose end points P', P" vary independently on R.
We denote the pair of points (P'P") by (7 r) and term P' and P" the vertices
of (71-). We shall regard pairs
(P'P"), (P"P0
as identical. With such a convention the ensemble of points (7 r) will be termed
the symmetric square of R. We denote it by P2.
We shall now show how R2 can be represented by a simplicial complex IT.
Suppose that R is the topological image of a simplicial m-complex K lying in a
^-dimensional euclidean space E. We represent P' and P" by their images on
K and the pairs (P'P") by the corresponding points on the product K X K.
We suppose K X K represented by a simplicial complex in the euclidean space
E X E, the product of E by itself. The complex K X K represents R X R.
To obtain a representation of R2 we proceed as follows.
Pairs (P'P") on R2 have been identified with pairs (Q'Q") for which
Q" = P',
Q' = P"
We denote this transformation by T. Points (ir) which are images of one
another under T will be termed congruent. Let (y 1, • • • , yM) = (y) be the co¬
ordinates of a point of E. If (yf) and (y") represent two points on E, the set
, f t » n.
(y \y * y y n y y 1 f * y y n)
can be regarded as representing a point on E X E. The transformation T will
be represented in the space E X E by a transformation of the form
(10.0)
r
Vi = Viy
y'i = y\ (i == 1 , • - - , n).
We shall prove the following.
(a). The complex K X K in the space EXE may be sectioned so as to have the
following properties. A cell which is not pointwise invariant under T will possess
THE CRITICAL SETS OF FUNCTIONS
182
[VI]
no pairs of points which are congruent under Ty and will he congruent under T
to a second unique cell of the complex.
We begin by sectioning K X K in the space E X E by each of the (2/i — 1)-
planes
(10.1) y'i = y] (i = 1, • • • , y).
After such a sectioning a cell which is not pointwise invariant under T will
possess no pairs of points which are congruent under T. For a given cell either
satisfies each of the conditions (10.1) identically or else for some value of iy say ky
satisfies one of the inequalities
(10.2) y'k < y"k, y'k > y"k
identically. If the cell satisfies the condition y[ < y"k) for example, it can possess
no pairs of points congruent under T, because under T a point satisfying yk < yk
is carried into a point satisfying yk < y'k.
With K X K so sectioned let w be a finite set of (2 m — l)-dimensional planes
in the space E X E so chosen that each fc-cell of K X X is on the A;-dimensional
intersection of a subset of these hyperplanes. Let w' be the set of hyperplanes w
together with their images under T. We now further section K X K by the
hyperplanes of w'. The resulting polyhedral complex will have the properties
required in (a).
To insure that the cells of K X K be simplices we further subdivide K X K in
the usual way by introducing a new vertex on each j- cell, j > 0, taking the cells
in the order of their dimensionality, and adding the “straight” cells determined
by the new vertices and the cells on the boundaries of the j-cell. We must take
care however to choose these new vertices on congruent cells as congruent points.
We thereby obtain a simplicial representation of K X K satisfying (a).
Finally we identify the pairs of congruent cells of K X K, obtaining thereby a
complex which we denote by n and which is the one-to-one image of the symmetric
square R2 of R.
The complex II could be represented as a simplicial complex on some auxiliary
space of sufficiently high dimensions. By the cells of R2 we mean the images on
R2 of the cells of n.
We shall not be concerned with the ordinary connectivities of R2y but rather
with certain “relative connectivities” of R 2 defined as follows. Cf. Lefschetz [1].
A point (ir) on R2 will be termed contracted if its vertices Pf and P" are identi¬
cal. A cell on R2 will be termed contracted if composed of contracted points (t).
In determining the relative boundaries of a fc-chain, contracted ( k — l)-cells
shall not be counted. With this understood relative cycles, homologies, and
connectivities (mod 2) are defined as are ordinary cycles, homologies, and
connectivities.
That these relative connectivities of R2 are finite can now be proved with the
aid of the simplicial representation n of R2y applying the Veblen-Alexander
deformation to each relative cycle ck (Lefschetz [1], p. 86). We observe first
til)
CRITICAL CHORDS OF MANIFOLDS
183
that if a point on a cell a 4 of P 2 is contracted, every point of is contracted.
This follows from the fact that K X K was sectioned by each of the (2/z — 1)-
planes (10. 1 ) . Recall also that under the Veblen-Alexander process a given point
of cii will be deformed through points all on the same closed cell of R 2. Hence
contracted points of at will be deformed through contracted points, and con¬
tracted cells through contracted cells. Thus each relative cycle of P 2 will be
relatively homologous to a cycle of cells of P2, and the relative connectivities of
P2 will be finite.
We note that any relative cycle of R 2 sufficiently near the set of contracted
points on P2 will be relatively homologous to zero. For any such relative cycle
Ck will be relatively homologous under the Veblen-Alexander process to a cycle
of contracted cells, and will thus be relatively homologous to zero.
Let ( 7r o ) be any point of R2 for which P' X P" . We shall represent the
neighborhood of (7 r0) as a Riemannian manifold. To that end let (u\ • • • , um)
be coordinates of a point on R in an admissible representation of R neighboring
P', and let (vl, • • , vw) be similar coordinates in an admissible representation of
R neighboring P". Let
g'lJduiduJ, (hkClPriP ( i,j , A, k = 1, • • ■ , m)
be respectively the corresponding differential forms of R. We now represent the
points on P2 neighboring (7 r0) by the local coordinates ( u ) and (v) combined, and
assign to these points (7 r) the differential form
(10.3) ds2 = g'ijdu'du3 + glkdvhdvk (?, j, hf k = 1, • • • , m).
We may thus regard R2 as a Riemannian form with a metric given by (10.3).
This statement is to be qualified by the remark that the neighborhoods of con¬
tracted points on P2 do not admit of a convenient parametric representation and
corresponding metric.
Critical chords of manifolds
11. We continue with the w-manifold P of the preceding section and its
symmetric square P2. If (7r) = {P'P") is a point on P2, the distance in the
euclidean space of P between the points P' and P" on R will be a function f(ir)
analytic in the local coordinates of P2 wherever these local coordinates have been
defined, that is, neighboring each point of R2 not a contracted point. Although
/ takes on its absolute minimum zero at contracted points on P2, such contracted
points cannot properly be included as critical points of /. This corresponds to
the use of relative connectivities instead of ordinary connectivities of P2.
One sees that a necessary and sufficient condition that / have a critical point
(7 r) is that the corresponding chord of P be a critical chord. I say, moreover,
that the lengths of critical chords of P are bounded away from zero for all such
chords of P. To verify this fact consider a normal to R at a point P. The
segment of this normal which consists of points at most a sufficiently small
positive distance L from P will have no point other than P in common with P.
184
THE CRITICAL SETS OF FUNCTIONS
[VI]
Moreover, one choice of the constant L can be made for the whole manifold R ,
so that all extremal chords must have lengths greater than L. We note the
following.
(a). If L is a positive constant less than the lengths of the critical chords of R,
the relative connectivities of the domain f < L on R2 are all zero.
For if e is any arbitrarily small positive constant less than L, one can readily
show, as in §2, that the domains f < e and/ < L are homeomoiphic. But if e is
sufficiently small, the points on f < e will be arbitrarily near contracted points,
and as we have already noted all relative cycles on f < e will then be relatively
homologous to zero on / < e. Hence statement (a) is true.
We must here ask whether the results of §§1 to 9 still hold when the con¬
nectivities there appearing are replaced by the relative connectivities of the
preceding section, and the convention is made that zero is not a critical value of
our function /. The answer is in the affirmative, but several explanatory remarks
are necessary.
In the first place each critical value c is positive, so that the spannable and
critical cycles neighboring the corresponding critical sets possess no contracted
cells. This part of the theory holds then exactly as before.
In carrying through the rest of the theory it is necessary that all deformations
have the property of deforming contracted points through contracted points.
The Veblen-Alexander deformation of a singular chain into a chain of cells of R2
has this property, as we have already remarked. The deformation D which we
have defined in §6 in what we have termed the Deformation Lemma, is not in the
present case defined for a contracted point. We can avoid the difficulties
inherent in this situation by altering D as follows. If L is any positive constant
less than the least critical value of /, we can perform D as defined in §6 at least
until the point deformed has reached a point at which / = L, thereafter holding
the point fast. Points for which / < L initially, are to be held fast throughout
the altered deformation.
With these changes the theory goes through as before until Theorem 7.3 is
reached. In proving this theorem for the case that a is less than the absolute
minimum of /, use was made of the properties of the absolute minimizing set.
In the present case these properties are replaced by the convention that / = 0
is not a critical value, and that the relative connectivities of the domain/ < L are
all zero.
Theorem 7,4 together with its corollaries then holds as before, relative con¬
nectivities replacing ordinary connectivities.
For present purposes it will be convenient to call any set of critical chords
which corresponds to a critical set of/, a critical set of chords , and to assign to such
critical sets of chords the type numbers of the corresponding critical sets. In
particular if a critical chord corresponds to a non-degenerate critical point, the
chord will be termed non-degenerate and assigned the corresponding index.
With this understood we restate Theorem 7.4 as follows.
[11]
CRITICAL CHORDS OF MANIFOLDS
185
Theorem 11.1. Between the sums Mi of the type numbers of the critical sets of
chords of R and the relative connectivities Ri of the symmetric square of R the relations
(1.1) still hold.
For the sake of a future reference we state the following corollary.
Corollary. If the critical chords of R are all non-degenerate , there exist at least
Ri such chords of index i.
We shall consider the critical chords of any analytic manifold homeomorphic
with an m-sphere. To that end we first consider the critical chords of the
ellipsoid Em
(11.1) a\x\ = 1 (i = 1, • • * , m + 1),
where
a i > a2 > • • • > am + i >0.
One sees that the only critical chords of Em are its axes. Concerning these
axes we shall prove the following lemma.
Lemma 11.1. yFhe axes of the ellipsoid Em form non-degenerate critical chords ,
which taken in the order of their lengths have indices given by the respective numbers
m, m + 1, • • ■ , 2m.
Let the symmetric square of Em be denoted by R2. Let the critical chord of
Em on the axis of xk be denoted by gky and the corresponding point (tt) on R 2 by
(71-*). Let the coordinates (x) of a point near the positive end of gk (xk > 0) be
denoted by (uif • • , um 4 1), and the coordinates (x) of a point near the negative
end of gk (xk < 0) be denoted by (iq, • • • , vm + i). We can represent / near the
point (wk) in terms of the parameters
ua, va (a = 1 , • • • , k — 1, k + 1 , • • ■ , m + 1).
This set of parameters can be regarded as an admissible set of coordinates on the
Riemannian manifold R 2 neighboring (tt*).
On Em near the positive end of gk we have
Uk = -1- (1 - alul)in (a = 1, • • • , k - 1, k + 1, • • • , m + I),
ak
(11.2)
— — (1 ~ |fl'aUa + ’ ’
ak
Similarly on Em near the negative end of gk we have
(11.3) Vk ■= — — (1 — h°lvl + • ’ ) (« = 1, • • • » k — 1, k + 1, • • • ,m + 1).
a*
186
THE CRITICAL SETS OF FUNCTIONS
[VI]
The length / of the chords determined by the parameters uay va will be given by
the formula
/ = \{ua “ Va)( XL a - Va) + ( Uk ~ Vk)2]U\
where a is to be summed as previously, but k not summed.
(11.2) and (11.3) we find that
f = j(wa - Va) (ua ~ Va) + [l ~ If (“» + +
for variables xia and va sufficiently near zero. Thus
9 f o o ^1/2
/ = Ok ( T” ^ U a ~~ Va^Ua ~ Va ^ ~ v ^ * * J *
Whence
(11.4) ^ = 1 + l 1 (a* - al)(ua ~ va)* - «2(m« + O2] + • • • •
Consider the following quadratic form in na and vay
Qa S [(<*J - al)(ua “ Va)2 ~ ««(^a + ?’a)2] (« 5* k)
with a and A* fixed. One sees that Qa is non-degenerate, and that its index is 1 or
2, according as aa is less than or greater than ak. It follows that the bracket in
(11.4) has the index
m + k — 1.
Making use of
J 2^1/2
In fact for k fixed, k — 1 of the m forms Qa have the index 2, and all of these
forms have an index at least 1. Now k runs from 1 to m + 1 so that these
indices run from m to 2m, and the lemma is proved.
We shall say that a critical chord is of increasing type if it corresponds to a
critical point of increasing type, in the sense of §7. In order to show that each of
the critical chords gk is of increasing type we shall show that there exists a linking
ju-cycle rM belonging to gk. The integer ^ is the index of gky that is,
V — m + k — 1,
as we have just seen.
To define I\ we shall subject the space (?) to a deformation in the form of a
rotation. In this deformation the time t shall increase from 0 to 1 inclusive. A
point whose coordinates (?) = (z) when t = 0 shall be replaced at each sub¬
sequent moment t by a point (?) such that
xp = Zg COS 7 rt — zp sin 7 rty
Xg = zp sin 7 rt + Zq COS irt
Xi = Zi,
(11.5)
(v t* 9; o < t g l).
[.11]
CRITICAL CHORDS OF MANIFOLDS
187
where p and q are two distinct, fixed integers on the range 1, • • • , m + 1, and i
takes on integral values from 1 to m + 1 inclusive, excluding p and q. When
t = 1/2 we note that
Xp Zqy
Xq = Zp.
The deformation rpq of points (x). Let (7 r) be any point on R 2. Let h be the
straight line in the space (x) which passes through the vertices of (x). Under
the deformation (11.5), h will be replaced at the time t by a straight line which
we denote by ht. If h is sufficiently near the origin, ht will intersect Etn in two
distinct points. We denote this pair of points by (x*). Under the deformation
rvq the point (x) shall be replaced by the point ( x, ) at the time ty 0 g t ^ 1.
The deformation rpq of (x) is defined only for points (x) for wrhich the cor¬
responding straight lines ht meet Em in tw o distinct points.
If Wk is a chain of points (x) on R2 for which rpq is defined, and if + i is the
deformation chain derived from wk , we shall wrrite
wk. fi = rpqwk.
Any point (7 r) on wk and the point (x') which replaces it under rpq when t = 1
will be identical on R 2, in accordance with our conventions. The chain wk and
the chain w'k which replaces wk under rpq when t = 1, will be identical on R2, and
will accordingly disappear from the boundary of wk+ l. It is clear that we can
then regard wk + 1 as the product of wk and a circle whose representative param¬
eter is t.
The chains H x on R2. We now consider the set of all chords parallel to the
extremal chord g L. The subset of these chords w^hose lengths are at least unity
will be determined by points (x) to which the deformation rvq will be applicable,
provided at least the semi-axes of Em are sufficiently near unity, and this we
suppose to be the case. The points (x) determined by these chords may be
regarded as points on a chain Hm on Rr . We suppose the semi-axes of Em are
so near unity that the deformation chain
Hm+k~ 1 = r*_i, k * • * rlzrl2Hin (k = 2, • • • , m + 1)
is well defined and possesses a boundary on w hich / is positive and less than the
length of pi. It is hereby understood that the deformations defining Hm+k- 1 are
not to be combined and then applied to Hmy but rather that each deformation is
to be applied to the chain which follows it to form a new chain of one higher
dimension. We note that
Hm+k- 1 = 7T-], kH m+fc-2 (* > 1).
Recalling that the extremal chord gk is determined by the point (x^.), we note
that (xfc_i) is replaced by (xfc) under rfc_ h k when t = 1/2. Starting with the
fact that (x 1) lies on Hm we then see that (xa,) lies on Hm+k- 1.
188
THE CRITICAL SETS OF FUNCTIONS
[VI.]
The relative cycles I\ on R 2. Let Bm+k- 2 be the boundary of Hm+k- 1. On
fim-f-jb-2,/ is less than the minimum critical value of/ on #2. Hence there exists a
relative bounding relation on R2 of the form
HL+k-i -» #-+*-2 (A: = 1, • * • , m + 1)
in which H„+k-1 is a chain below the length of g\. With the chord gk we now
associate the relative cycle
] = Hm+k— 1 ~f~ Hm+k — i •
We shall presently show that rm+jt_i is a linking cycle belonging to gk.
We shall first prove the following lemma.
Lemma 11.2. The relative cycle Fm is a linking cycle on R2 belonging to the critical
chord gi.
Let (7 r) be a point on R 2 neighboring the point (71-1) which determines g i.
Let P' and P" be the respective vertices of (71-) neighboring the ends of gi at
which Xi is positive and negative. Neighboring (wi), R 2 can be admissibly
represented in terms of the last m coordinates (x) of P' and P" respectively.
Denote these coordinates by
(11.6) x'a, xQ ( a = 2, • • • , m + 1),
respectively. To represent Tm regularly neighboring (7r0 on R2 it will be suffi¬
cient to represent Tm regularly in the space of the parameters (11.6). Such a
representation can be obtained by setting
u a (a — 2 , ■ ■ ■ , m + 1),
and assigning the variables ua independent values near 0.
On the value of/ at the point (71-) determined by the variables ua will be an
analytic function
/ == <p(U2f f Um + 1)
of the variables ua for variables ua near 0. Moreover the function <p has a proper
maximum when these variables are null, at the point (wj). We can accordingly
apply Theorem 7.5, and infer that there exists a positive constant e so small that
the locus,
if(u) = *>(0) - e,
on R2} will be a spannable ( m — l)-cycle Cm_i belonging to the function / and its
critical point (71-]). But if e is sufficiently small, this cycle bounds on R2 below
<p( 0), in fact bounds the domain
/ ^ *(0) - e.
Thus rm is a linking cycle belonging to (71-1) and the extremal chord gi.
(11.7)
***a
tt
[11]
CRITICAL CHORDS OF MANIFOLDS
189
We shall now introduce a principle of use in the representation of deformations.
Let Mr be an r-dimensional manifold in a space of coordinates (x) admitting a
representation
A = h%(v i, • • • , vr) (i = 1, • • * , q)
in terms of r parameters (v) neighboring a point (vQ) . We suppose that the param¬
eter values (?;0) determine (z0), and that neighboring (v0) the functions hx(v)
are of class Cl. Let D be a deformation in the space (r) in which t represents
the time, and in which a point whose coordinates xt assume values zx when
t — t0} is replaced at the time t, for t neighboring to, by the point
xx = Xi(z, t) (i = 1, • ■ * , q),
where the functions xt(c, t) are of class Cl in (z) and t, for sets (z) near (x0) and t
near t0 , and
Zt SE Xi(z, to).
For sets (z) near (xo) and t near to, the equations
(11.8) xx = xx[h(v), t] (i = 1, • • • , q)
will define an (r + 1 )-dimensional manifold which will be termed the deformation
manifold Mr+ 1, corresponding to Mr and D. The representation of Mr + 1 in
terms of the parameters ( v ) and t will be termed the corresponding product
representation , We now state a principle of use in the sequel.
Composition Principle. A sufficient condition that the product representation
of the deformation manifold Mr+i in terms of the parameters (i v ) and t, be regular
when {v) = ( vo ) and t — to , is that the parameters ( v ) regularly represent the mani¬
fold Mr neighboring ( Vo ), and that the trajectory of the point ( v ) = (r0) under D be
not tangent to Mr when t = £0.
This statement follows at once from the representation (11.8) of M r+i-
To show that rm+*_i is a linking cycle belonging to gk we need to know that this
cycle admits a regular parametric representation on the Riemannian manifold
R 2 neighboring (t*). Recall that
Hm+k- 1 = rk-l, kHm+k- 2 (k > 1).
Also recall that R 2 admits coordinates (11.6) neighboring (xi), and that in the
space of these coordinates (11.6), Hm admits the regular representation (11.7),
and thus a regular representation on R2. Proceeding inductively we shall
assume that Hm+h~ 2 admits a regular representation on R2 neighboring (7r*_i)
in terms of parameters (p) neighboring (v0)* We then come to the following
lemma.
Lemma 11.3. The points on Hm+k-\ neighboring (t*) result from the deformation
of Hm+k-2 neighboring (tt*_i) under rk- 1, h,for values of t which neighbor 1/2. If
190
THE CRITICAL SETS OF FUNCTIONS
[VI]
Hm+k- 2 admits a regular representation neighboring (ir k~i), in terms of parameters
(t>) neighboring (t»o), the corresponding product representation of Hm+k- 1, in terms
of the parameters ( v ) and t, will be regular when (v) = (u0) and t — 1/2, provided
the semi-axes of Em are sufficiently near unity.
Let (t) be a point on R 2 neighboring (tt*). Let P' and P" be the vertices of
(t) neighboring the vertices of (tt*) at which x* is positive and negative re¬
spectively. Let
(11.9) x'a, x' (a = 1, • • * , k - 1, A: + 1, • • , m + 1)
be the ^-coordinates of P' and P" respectively, omitting the /cth coordinates.
The coordinates (11.9) will serve as Riemannian coordinates of R 2 neighboring
(»*)•
Let Cm+k~2 be the chain on //m4 k~i into which Hm+ *_2 is deformed under
rk-],k when t = 1/2. The point (wk) is on Cmu- 2, and neighboring (717) the
chain (7w4*~2 is regularly represented })y the parameters (?>) which represent the
corresponding points on Hm^k- 2. This is true if the semi-axes of #m are unity.
It is then also true if the semi-axes are sufficiently near unity. We shall now
apply the Composition Principle to complete the proof of the lemma.
To that end we note that the chords determined by //m4; 2 are parallel to
the ( k — l)-plane of the xb x2, ■ • • , Xk- 1 axes in the space (x). Hence the
chords determined by C„+*_ 2 are parallel to the (k — l)-plane of the
•r 1 , * > _2 , J k
axes. It follows that on Om+*_2,
(11.10) dx[^l = dx^.
On the other hand consider the trajectory 7 traced by (71-* _ 1) under r*_i, This
trajectory passes through (7r*) when t = 1/2. On it the vertices of the points
(7r) are symmetrically placed relative to the origin in the space (x). In partic¬
ular on 7 at (71**),
(11.11) = -"-ft-1 0.
at at
A comparison of (11.10) and (11.11) showrs that 7 is not tangent to Cm+k-2 at
(717). The lemma follows from the Composition Principle.
We can now prove the following theorem.
Theorem 11.2. The relative cycle rm4.*„A is a linking cycle belonging to the
critical chord gk , provided the semi-axes of Em are sufficiently near unity.
The proof of this theorem is similar to the proof of Lemma 11.2. It makes
use of Theorem 7.5 and depends upon a preliminary verification of three facts.
I. The index of gk is m + k — 1.
II. The cycle rm+*_i admits a regular representation neighboring (7^).
Ill]
CRITICAL CHORDS OF MANIFOLDS
191
III. On H m+&-i the chord length/ assumes an absolute proper maximum at
(*■*)•
Statements I and II have already been established. We turn therefore to
III.
Recall that the chords determined by Hm+k- 1 are parallel to the A>plane X*
determined by the Xi, • • • , xk axes. On \k the chord of maximum length is
gk. But all other A>planes parallel to X* which are not tangent to Em either fail
to intersect Em or intersect Em in an ellipsoid similar to the ellipsoid
a\x\ + • * • + a\x\ = 1,
but with semi-axes which are shorter. The chords on these ellipsoids are all
shorter than gk. Statement III is accordingly proved.
As in the proof of Lemma 11.2 we now turn to Theorem 7.5. We infer that
the locus
/ ~ / 00 = -e
on rm+*_i, for a sufficiently small positive constant e, is a spannable (m + k — 2)-
cycle associated with (71-*), and that this spannable cycle bounds below f(irk)-
The cycle is accordingly a linking cycle associated with gkj and the
theorem is proved.
From the fact that the axes of Em are its only critical chords, and that each of
these chords is of increasing type, we deduce the following theorem. See
Corollary 7.4 and Remark.
Theorem 11.3. The relative connectivities Ri of the symmetric square of the
m-sphere are all zero except that
Rfn Rm-\-l * Rirn L
These results were obtained by the author in 1929 (Morse [7]). More recently
M. Richardson and P. A. Smith [1] have taken up the abstract topology of invo¬
lutions including the topology of symmetric products, and have obtained
important general results. If suitably modified for the case of relative con¬
nectivities the theorems of Smith and Richardson joined with the above theory
will greatly enlarge the results on critical chords.
We state the following corollary of the theorem.
Corollary. If R is any regular analytic image of an m-sphere whose extremal
chords are non-degenerate y among these extremal chords there must exist m \ ex¬
tremal chords with indices varying from m to 2m inclusive.
In the degenerate case the same result holds provided each critical set of chords
is counted according to its type numbers.
CHAPTER VII
THE BOUNDARY I ROBLEM IN THE LARGE
The problem of extending the theory of critical points of functions to the
theory of critical points of functionals presents new analytical and topological
difficulties. The point is replaced by a curve whose end points satisfy the given
boundary conditions. Three types of curves are used: the ordinary continuous
curve, the curve of class Dl> and the broken extremal. The broken extremal is
represented by the ensemble (x) of its end points and vertices. The continuous
curve is used to give the topological part of the theory a purely topological basis.
The broken extremal and points (V) are used to approximate the functional by a
function and the curves of class I)1 serve to mediate between the continuous
curves and the broken extremals.
The local characterization of the critical sets of the function J(tt) is made
difficult by virtue of the fact that the critical sets are at least y>-dimensional,
where p is the number of intermediate vertices in a point (t). In general these
critical sets are open. These difficulties are surmounted largely with the aid of
J -normal points (x), that is, points (7 r) which determine successive extremal
arcs on which J has the same value.
The ensemble of continuous curves which satisfy the given boundary condi¬
tions form the basic space 12. In general the space il has infinitely many con¬
nectivity numbers which are not null. To parallel the work of the preceding
chapter and obtain relations between the connectivities of 12 and the type
numbers of the critical sets of extremals requires a careful use of deformations.
These deformations have two essential characteristics: they are invariant in
character, that is, locally independent of the coordinate systems used, and when
applied to curves of class Dl do not increase the value of J. In this connection
we find it necessary to introduce a new definition of the distance between two
curves of ft of class D1. This distance possesses two important properties. It is
invariant in character, and by virtue of it J may be regarded as a continuous
functional.
In the final section of this chapter we apply the preceding results to prove the
existence of infinitely many extremals joining any two points on the regular
analytic homeomorph of an m-sphere, including thereby a characterization of
these extremals.
The calculus of variations in the large was first studied in connection with the
absolute minimum. Hilbert was a pioneer in this research. See Bolza [1],
p. 428. Tonelli [1] has added many important new conceptions and theorems.
Signorini [1] and Birkhoff [1] have effectively used the broken extremal. Mc-
Shane [1] has extended Tonelli’s work. Carath^odory [4] has recently studied
the general positive regular problem and obtained novel results. The author is
192
THE FUNCTIONAL DOMAIN 12
193
concerned with the minimizimg extremal only as one type of critical extremal
and has made little use of the theories of the absolute minimum.
The reader may also refer to a paper by Richmond [2] in which theorems in the
large depending upon the existence of a field of extremals are obtained.
The functional domain £2
1 . We are concerned here with the Riemannian space R of Ch. VI. The space
R is the homeomorph of an auxiliary simplicial m-circuit K . Locally it possesses
an analytic Riemannian metric as described in Ch. VI.
Let A1 and A2 be any two distinct points on R. The points A1 and A2 will be
used to designate the initial and final end points, respectively, of an admissible
curve on R. Let B r be an auxiliary connected simplicial r-circuit for which 0 ^
r < 2m. Let Z be a set of pairs of distinct points (A1, A2) on R, such that the
pairs (A1, A2) are homeomorphic with the points of Br. On R the local co¬
ordinates (x) of A1 and A2 will be respectively denoted by
(x11, • • • , xml), (x12, • • * , xm2).
If r = 0, the end points A1 and A2 are fixed. If r > 0, we suppose that the
neighborhood of each point of Br can be represented as the image of a neighbor¬
hood of a point (a0) in an auxiliary euclidean r-space of coordinates (a), and
that the corresponding points (A1, A2) can be locally represented in the form
x“ = xf*(a) (i = 1, * • • , m\ s = 1,2),
where the functions xiJ,(a) are analytic in (a) for (a) near (a0), and possess a
functional matrix of rank r. We call this set of pairs of points (A1, A2), the
terminal manifold Z .
Let 7 be the continuous image on R of a line segment 0 ^ I ^ 1. If the end
points t — 0 and t — 1 of 7 determine a pair (A1, A2) on Z, 7 will be termed
topologically admissible. When we are dealing with our integral on R we shall
suppose that 7 is of class Dl as well as topologically admissible. We then term
7 a restricted curve. Evaluated on restricted curves the integral defines our
basic functional J.
The totality of topologically admissible curves 7 will be termed the functional
domain £2 determined by R and the manifold Z.
We shall now define chains and cycles on £2.
Let closures be indicated by adding bars. Let at be any t-simplex in an
auxiliary euclidean space aDd p a point on at. Let t be a point on the interval
0 g t ^ 1. Denote this interval by h. The pairs (p, 0 make up a product
domain on X t\. Let p represent a continuous map of on X t\ on R. The image
under <p of the product p X U will be called the curve determined by p. We
suppose each such curve is topologically admissible. In such a case the image of
X ti on R will be called an i-cell a» on £2. If a* is any fc-simplex on the bound¬
ary of a*, the image of a* X U under <p will be called a boundary k-cell of a».
194
THE BOUNDARY PROBLEM IN THE LARGE
[ VII ]
It will be convenient to refer to a product domain such as a ; X G as a func-
t io rial i-simplex .
Let ft be an auxiliary /b-simplex. We do not exclude the case where a{ ~ ft.
Suppose that k S i and that T represents an affine projective correspondence
wffiich maps a* onto ft, covering each point of ft at least once. Let p be any point
on cti and q its image on ft. If the points ( p, t) and ( q , l) on a{ X t\ and ft X tv
respectively are now regarded as corresponding, there results an affine projective
map of oci X t\ onto ft X U, covering each point of ft X t\ at least once. Such a
map of a, X L on ft X t\ will be termed an admissible affine map of X U
on ft X t.
Let \f/ represent a continuous map of ft X t\ on R. Suppose that for some
admissible affine map of X U on ft X G, corresponding points have the same
images on R under <p and \p respectively. If i — k, the images of a* X t and
fii X ton R will then be regarded as identical /-cells on SI. If i > k , the image of
a% X tv on R will be counted as a null /-cell on S2.
By a dosed 2-cell on SI we mean an /-cell on S2 together with its boundary cells
on SI. By an i-chain on 12 we mean a finite set (possibly null) of closed /-cells on
12, no two of which are “identical.” By the sum, mod 2, of tw o /-chains Zi and
Wi on 12 w'e mean the set of closed 2-cells which belong to Zi or to uu> but not to
both Zi and uu. The boundary z%..x of an /-chain Zi on 12 is defined as the sum,
mod 2, of the closed (i — l)-cells which are “boundary cells” of /-cells of
We then write
(1.1) 1.
As previously a A:-chain on SI whose boundary is null is called a k-cycle on SI.
Homologies, independence, maximal sets of A;-cycles on 12, are now formally
defined for the case of 12 as for the case of R. We note in particular that ct_i in
(1.1) is now an (/ — l)-cycle on 12. We then w rite
Zi- 1 ~ 0 (on 12)
as before. If for /-cycles at and ft on 12, + ft ^ 0, we also write a, ^ bi. It is
then clear that the respective members of valid homologies or bounding relations
such as (1.1) can be added mod 2.
Let ai be an /-cell on 12 given as the image on R of a functional simplex ai X t\
under a map <p. To subdivide a/we first subdivide at-. Let pi be any one of the
resulting simplices. We replace ft by the sum of the images under <p of the
closed functional simplices Pi X t\. To subdivide a chain on 12 we subdivide its
cells in the order of dimensionality. By the connectivity P / of 12, j = 0, 1, • • • , we
mean the maximum number of j-cycles on 12 between which there is no homology,
provided such a maximum exists. If no such maximum exists, we say that P,
is infinite. A necessary and sufficient condition that P0 = 1 is that any two
admissible curves 7 be continuously deformable into each other among admissible
curves 7. The number P0 can be infinite. It will be infin’te, for example, if
wTe are dealing with curves joining twTo fixed points on a torus. In the case of
THE FUNCTIONAL DOMAIN il
195
curves on an ra-sphere (m > 1), with ends fixed, we see that 1\ = 1, but we shall
find that infinitely many of the remaining connectivities are not null. The
same is true for curves on an w-sphere one of whose end points is fixed and the
other of which is free to move on a ^-manifold with k < rn.
The connectivities of ft are invariant under any topological transformation of
R which carries admissible end points ( A l, A2) into admissible end points. For
purposes of pure topology the analyticity of R and Z is of course unessential.
Deformations on ft. The determination of the connectivities of ft and the
relations of these connectivities to our functional J lead us to deformations of
curves and chains on ft. In ordinary topology deformations of chains may not
necessarily be point deformations, that is, the deformation of a point may depend
upon the cell on which the point is given. So here the broadest class of deforma¬
tions, namely deformations of chains on ft, will not in general be curve deforma¬
tions, that is, will not be uniquely determined when a curve of ft is given, but
only when the curve is given on a cell of some chain of ft. To define such de¬
formations we proceed as follows.
Let ai be an 2-cell on ft, the image on R of a functional simplex o; X h. Let p
be any point on aiy and t and r be points on the respective intervals
O^gl, 0 g r ^ 1,
denoted by h and rx respectively. The sets (p, /, r) represent points on the
product
(1.2) X h X rj.
Let <p(p, t , r) represent a point on R which is the continuous image on R of an
arbitrary point (p, ty r) on (1.2). Suppose moreover that the points
<p(p, r)> <p(p> T)
form an admissible pair of end points (A1, A2), and that when r = 0, <p(py ty 0)
is the map which defines a». We say then that <p(py tf r) defines a deformation
D of ai on ft. Under D a point <^(p, /, 0) on a* is said to be replaced at the time
r by the point v?(p, t , r).
Let be a simplex which is the affine projective image of a,. The domains
(1.3) XI] X Tlf X h X rj
then admit an affine projective correspondence in which points (p, t, r) on
ai X t\ X ti correspond to points ( q , ty r) on X h X n whenever p and g
correspond on ai and Pi respectively. This affine correspondence between the
domains (1.3) will be termed admissible.
Let (p and \p now represent continuous maps of the respective domains (1.3)
on R of such a nature that the maps
<p(p, ^ 0), \f/(qy ty 0)
196
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
define the same i-cells at of 12, and *p and ^ define deformations of a,- on Q. When
the domains (1.3) possess an admissible affine correspondence by virtue of which
the maps <p and \p on R of projectively corresponding points on the products
(L3) are identical, the maps <p and \f/ will be said to define the same deformation
of a, on 12.
To deform a chain on 12 one deforms its cells, admitting however only such
deformations as replace conventionally identical cells by cells which may be
regarded as identical. As in ordinary topology one proves that two cycles z< and
Wi which can be deformed into one another on 12 bound a chain on 12.
The function J(ir)
2. In terms of any set of local coordinates (r) of R and of variables (r) ^ (0)
we suppose that the invariant function F(x , r) of Ch. V is here analytic, positive,
and positive regular. Moreover F(x, r) shall be homogeneous of order 1 in the
variables (r).
As is well known there then exists a positive constant e , small enough to have
the following properties. Any extremal arc E on wffiich J < e will give an
absolute minimum to J relative to all sensed curves of class Dl joining E’& end
points. On E the local coordinates of any point will be analytic functions of the
local coordinates of the end points of E and of the distance of P along E from
the initial end point Q of E , at least as long as E does not reduce to a point.
The set of all extremal segments issuing from Q with J < e will form a field
covering a neighborhood of Q in a one-to-one manner, Q alone excepted. We
nowr choose a positive constant p less than e, and make the following definition.
Any extremal segment on R for which J is at most p will be called an elementary
extremal.
An ordered set of j) + 2 points
(2.1) A\ P\ • • • , Pp, A2
on R, with (A1, A2) on the terminal manifold Z, will be denoted by (r). The
points (2.1) will be called the vertices of (tt). It may be possible to join the
successive points in (2.1) by elementary extremals. In such a case (t) will be
termed admissible . The resulting broken extremal will be denoted by g(rr)f
and also termed admissible. The value of J taken along g( tt) will be denoted
by J(tt).
We can regard J (7r) as a function <p of the parameters (a) locally representing
its vertices A1, A2 and of the successive sets of coordinates (x) locally represent¬
ing its vertices PJ. The function <p will be analytic, at least as long as the
successive vertices remain distinct. A point (w) whose successive vertices are
distinct will be called a critical point of J( w) if all of the first partial derivatives
of the function <p are zero at that point.
If the successive vertices are distinct, the conditions that <pah be null are seen
to be
[>„(*, - o
(2.2)
(h = 1, • • • , r).
[2]
THE FUNCTION J(t)
197
Here ( x , x) is to be evaluated on g( t) at the final end point of g( n) when s = 2,
and at the initial end point of g( tt) when 5=1. The r conditions (2.2) for
h = 1, • • • , r and r > 0 are equivalent to the transversality conditions of Ch.
V, §9.
The partial derivative of <p with respect to the zth coordinate of a vertex (x)
is seen to be
(2.3) Frt(x, p ) - Frl(x, q),
where (p) and (q) are the direction cosines at (x) of the elementary extremals of
<7(71*) preceding and following (x) respectively. If the difference (2.3) vanishes
for i — 1, • • • , m, I say that (p) = (q). For if these differences all vanish we
have
(2.4) P ) - V'FrA.x, q) = F(t, p) - piFTi(r , 7) = E(jt, q, p) = 0,
where E(x, py q) is the Weierstrass ^-function. Hut by virtue of the positive
regularity of F , (2.4) implies that (p) = ( q ) as stated. We conclude that a
necessary and sufficient condition that an admissible point (jr) whose successive
vertices are distinct be a critical point, is that g(ir) be a critical extremal, that is,
one satisfying the transversality conditions (2.2).
By a critical set ar of J(w) we mean any set of critical points on which J(t) is
constant, and which is at a positive distance from other critical points of J( t).
A critical set need not be connected. It is not necessarily closed, since it may
have limit points (71-) whose successive vertices are not all distinct.
To analyse sets of critical points (tt) we need to formulate the analytic condi¬
tions that an extremal neighboring a given critical extremal g be a critical
extremal. To that end let P0 be a particular point of g and (x) a set of
local coordinates neighboring P0. The extremals neighboring g with directions
neighboring those of g at P0 can be represented in the coordinate system (x) in
terms of the arc length t and 2 (m — 1) parameters (£) neighboring a set (£0) deter¬
mining g. These extremals take the form
(2.5) x{ = x{(t} P)
where the functions x*(£, p) are analytic in their arguments. In any other
coordinate system (x) representing the neighborhood of any other point of g,
the extremals with points and directions sufficiently near a point and direction
of g can again be represented in the form (2.5). We understand that the param¬
eters (fi) assigned to an extremal E in this second representation are the same
as those which belong to E’s continuation, cf. Ch. V, §5, in the first representa¬
tion, and that the arc length t in the second representation is measured from the
same point on the extremal E as in the first representation.
The conditions on the end points of an admissible arc have been locally given in
terms of parameters (a), but these parameters (a) can be eliminated and the
THE BOUNDARY PROBLEM IN THE LARGE
198
[VII]
conditions on end points (x1), (x2) neighboring the end points of g given in the
form
(2.6) \^(xn, * • • , xml, x12, - * * , xm2) = 0 (q = 1, • • • , 2m — r)
where the functions \pq are analytic in their arguments for points (x1) and (x2)
near the end points of g, and possess a functional matrix of rank 2 m — r. In
terms of the parameters t and (0) of (2.5) conditions (2.6) take the form
(2.7) Av(t \ t\ 0) = 0,
where tl and t 2 are the unknown end values of t, and the functions Aq are analytic
in their arguments for (0) near (0O) and t1 and t2 near the values which determine
the end points of g. To the conditions (2.7) must be added the transversality
conditions. In terms of the parameters (0) and the end values tl and f of the are
length ty these conditions take the form
(2.8) Bh(t\ t 2, 0) = 0 (h = 1, ••• , r),
where the functions Bh are again analytic for (0) near ( 0O ) and tl and i2 near the
values which determine the end points of g.
The conditions (2.7) and (2.8) are the required conditions that an extremal ( 0 )
near g be a critical extremal. These conditions may have no real solution other
than the initial solution (tfj, tl, 0o) corresponding to the given critical extremal g.
If this is not the case the real solutions of (2.7) and (2.8) neighboring the initial
solution will be representable by means of functions “in general” analytic on
one or more “Gebilde” (Osgood [1], lyoopman [1] with Brown) of p independent
variables, with p > 0, each G including the initial solution. Moreover any real
solution (tl, t2, 0) neighboring the initial solution can be connected to the initial
solution among real solutions of the form
tl = *l(r),
(2.9) t 2 - t\r)y
0i = Hr) tf = 1, ,2 (m - 1)],
where the functions on the right are analytic in r for 0 S r ^ 1, except for at most
a finite set of values at which the functions are continuous at least.
We now evaluate J along the extremal Er determine by the parameters 0i =
0i(r) in (2.9), taking J between the points determined by tl = tl(r) and t2 = t2{r)
respectively. The integral J then becomes a function J(t). One can simplify
the integral w bich gives J (r) by making a linear transformation from the arc
length t to a parameter u which varies between 0 and 1. By virtue of the fact
that each extremal Er satisfies the Euler and transversality conditions one sees
that J'{t) = 0.
We conclude that J is constant on the critical extremals neighboring g .
We shall regard a family of critical extremals as connected if any curve of the
family can be continuously deformed into any other curve of the family through
[2]
THE FUNCTION ./(tt)
199
curves of the family . By virtue of the preceding analysis in the small we see that
J is constant on any connected critical family. We shall continue with a proof
of the following statement.
(A). The critical extremals on which J < 6, a constant , can he grouped into a
finite set of connected families.
If the contrary were true there would exist an infinite set
(2.10) EuEt, •••
of critical extremals on each of which J < b, no two of which could be connected
among critical extremals on which J < h. Let l\ be a point on E{. The points
Pi will have at least one cluster point P. Let
(2.11) Qi,02,
be a subsequence of the points P% tending towards P as the index n of Qn becomes
infinite. Let (x) be a local coordinate system neighboring P. In the coordinate
system (x) let (a)n be the set of direction cosines at Qn of that extremal of the
set (2.10) on which Qn lies. The sets (a)n will have at least one cluster set (a).
The extremal E passing through the point P with direction cosines (a) will be a
critical extremal on which J ^ b . But as we have seen in the preceding para¬
graphs, E will be connected to all critical extremals defined by points Qn and sets
(a)n sufficiently near P and (a) respectively.
From this contradiction we infer the truth of statement (A).
Let g be a sensed curve of class IP and y a curve segment on g. The value of J
taken along y in the positive sense of g will be termed the J -length of y on g.
Let Po be the initial point of g and P an arbitrary point of g. The value of J
taken along g from P0 to P will be termed the J -coordinate of P on g. If the
J -coordinate of P is a differentiable function h(t) of the time t. P will be said to
be moving on g at a J-rate equal to | h'(t) |.
We shall now prove the following lemma.
Lemma 2.1. A mong points (x) for which J (tt) is less than a constant b, and for
which (p + 1 )p > b, there is at most a finite number of distinct connected critical sets.
Let a point (7r) such that the elementary extremals of g( tt) have equal J-lengths
be termed J -normal. Let // be any connected family of critical extremals on
which J < b. We observe that the set of J-normal crit ical points (71-) determined
by the extremals of H will form a connected set of points (tt). From statement
(A) we can infer that all ./-normal critical points (tt) for which J( w) < b can be
connected among such critical points to a finite set of such points (7 r).
We now pass to the case where (7r0) is any critical point for which J( tt0) < b.
The point (7r0) will not in general be J-normal. Let (71- j) be the J-normal critical
point which determines <7(7^). The point (7r0) can be connected to (71- j) among
admissible critical points (t). To that end we let each vertex of (7 r0) move along
0(7rn) to the corresponding vertex of (71-1), moving at a J-rate equal to the
“J-length on <7(7r0)” of the arc to be traversed. At the end of a unit of time
200
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
the point (tt0) will coincide with (ti). The point (to) is thereby connected
among critical points (t) to the /-normal critical point (irj).
The lemma follows from the results of the preceding paragraph.
The domain /( t) < b
3. Let b be an ordinary value of /. Suppose the number (p + 2) of vertices
in (t ) is fixed and such that
(3.1) (p + 1 )P > 6,
where p is the constant chosen in §2. Understanding that /(t) is defined only
for admissible points (t) we come to the problem of proving that the connectivi¬
ties of the domain /(t) < b are finite. We shall accomplish this with the aid
of certain deformations which we term J -deformations. These deformations
deform admissible points (t r) through admissible points (t). They do not
increase /(t) beyond its initial value, and they deform chains of points (t)
continuously. They are invariantive in their definition.
The deformation D' . Let (to) be an admissible point (t). As the time t
increases from 0 to 1 let the p intermediate vertices Pl of (t) move along g(irQ)
from their initial positions on ^(t0) to a set of positions on ^(t0) which divide
g(ir0) into p + 1 successive arcs of equal /-length, each vertex moving at a
/-rate equal to the /-length on g of the arc of g(-n0) to be traversed.
The deformation D' thereby defined is a /-deformation. In fact under Z)',
/ (t ) never exceeds its initial value /(t0) by virtue of the minimizing properties
of elementary extremals. Moreover, during Df the /-length of each segment
of g (t o) between two successive moving vertices varies between its initial value
and its final value /(t0)/(p + 1). But /(t0) < b, and upon using (3.1) we see that
J(*o) b ^
p 4- 1 p + 1 Pj
so that the corresponding elementary extremal never exceeds p in /-length.
Finally chains of admissible points (t) are clearly deformed continuously under
D'. Thus Df is a /-deformation.
The deformation D' tends to equalize the /-lengths of the elementary ex¬
tremals of g (t ) . The deformation D n now to be defined tends to lessen /( r)
when g(ir) has comers, or is an extremal but not a critical extremal.
To define Dn we need certain general facts relating to the possibility of assign¬
ing Riemannian metrics to the products of Riemannian Spaces or their subspaces.
Let R i, • • • , R2 be Riemannian m-spaces of which Rk possesses the local co¬
ordinates xl, i = 1, • • • , m, and an element of arc dsk such that
(3.3) dsl = giijdxidx’k (i, j = 1 , ■■■ ,m)
where k is not summed. The combined set of coordinates
[3]
THE DOMAIN J( ir) < b
201
will represent a point on the product 2 of the spaces Hi, ■ • • , Rq. To 2 we can
assign a metric defined by the form
ds2 = gktidx'kdx’k,
where k is now to be summed as well as i and j.
On the other hand let A be a Riemannian y-space with local coordinates ( z ).
Let B be a regular subspace of A, that is, a subset of points of A locally repre¬
sentable in the form
** = zi(u\ * • * , (i = 1, • • • ,v),
where the functions z'(u) are analytic in the variables ( u ) neighboring a set (u0)
and possess a functional matrix of rank p. If A possesses a metric defined by
the form
ds2 = Qij(z)dzidzJ (t, j = 1, •
we understand that the corresponding metric of B is defined by the form
ds2 = gi\z{u)]^d£duW,
' * > v)t
or, more concisely,
ds* = bhk(u)duhduk (h, k = 1, • • • , yu).
We regard the parameters (it) as the local coordinates of B.
With this understood we consider the (p + 2)-fold product A of i? by itself,
represent ing a point on .4 by the local coordinates on R of the points
A\P\ ... , P\A*
previously used to define vertices of a point (7 r). We can assign a metric to A in
the manner just described. To obtain admissible points (?r) one must limit the
pairs (A1, A2) to pairs on our terminal manifold Z. With the vertices A1, A2 so
limited, the corresponding point (ir) defines a point on a regular subspace B of A
Let (u) be a set of /i = r + pin variables of which the first r are the parameters
(a) used in a local representation of the terminal manifold Z, the next m are local
coordinates of Pl, the next m are local coordinates of P2, and so on, the last m
being the local coordinates of Pp. The complete set (u) forms a set of local co¬
ordinates in a representation of the regular subspace B of A . As in the preced¬
ing paragraph we can make use of the metric of A to derive a corresponding
metric
(3.4) ds 2 = bhk(a)duhduk (h, k = 1, • • * , p)
for the space B.
If any two successive vertices of a point (7r) on B are at a J-distance at most p
from each other, the point (t) on B will be admissible. The totality of admis¬
sible points (tt) on B will be denoted by IT. If (7 r0) is an inner point of II, J(w)
202
THE BOUNDARY PROBLEM IN THE LARGE
[ VII ]
will be defined for neighboring points (7 r) on II. The points on II neighboring
(tt 0) can be represented as above in terms of p = r + pm parameters (u). In
terms of these parameters (1 u ) we then set
(3.5) J( tt) =
obtaining thereby an analytic representation of J( tt) neighboring (7r0).
The set 2 and constant 77. In the forthcoming definition of the deformation
D" we shall refer to the set of all J -normal points (tt) on the domain J ^ h as
the set 2. We shall also refer to a positive constant 77 defined as follows. The
constant ?? shall be a positive constant so small that any point (71-) on B within a
geodesic distance 77 on B of points of 2 will possess successive vertices which are
distinct and define elementary extremals of J-length less than p. That such a
choice is possible follows from the fact that points (71- ) on 2 determine elemen¬
tary extremals with lengths which are uniformly bounded from zero and which
are at most the constant
The deformation D" . With this choice of 77 let (7r0) be a point on II within a
distance 77 of a point of 2. Neighboring (irf) we regard II as a Riemannian
manifold with metric defined by (3.4), and with parameters (u) in terms of which
J(w) equals the function <p(u) of (3.5). If (7r0) is an ordinary point of J{ ir), the
trajectories on n orthogonal to the loci on wrhieh J ( w) is constant, can be locally
represented by differential equations of the form
(3.6) ~ = - b»(u)<pul(u) ( i,j = 1, ,n),
where hij is the cofactor of the coefficient hi} in (3.4) divided by the determinant
| bij |. On these trajectories neighboring an ordinary point (7r0) of <p(u) we have
(3.7)
dJ d<p
dt dt
< 0.
Under the deformation Z)" points (ir) on 11 which are initially at a distance d
77/2 from the points of 2 shall be held fast. A point (71*) which is at a distance
d <77/2 from 2 shall be replaced at the time t, 0 g r | 1, by the point on the
trajectory (3.6) through (ir) at which t is larger than at (71-) by the amount
«(n/2 ~ d)r.
Here e is a positive constant which we choose so small that the points initially
at a distance d < rj/2 from 2 are deformed under D" through points at most a
distance 77 from 2. This choice of e is made in order that the points initially
at a distance d < 77/2 from 2 may be deformed through points at which the
corresponding function <p(u) never fails to be analytic through coalescence of
some of the vertices of (t).
[3]
THE DOMAIN J (tt) < b
203
The deformation Dp. The preceding deformations D ' and Z)" will now be
combined into the product deformation
(3.8) Dv = D"D\
It is understood that D' is applied first and D" then applied to the resulting
points. The subscript p indicates that we are dealing with points (t) with p
intermediate vertices. Concerning Dp we now prove the following lemma.
Lemma 3.1. V nder the deformation l)v each ordinary point on the domain
J{ tt) g b
is carried into a point (-n') at which < ,/( tt).
To prove this lemma we divide points (7r) on J (j) ^ b into two classes as
follows.
Class I shall contain the points (tt) which are deformed under D' into points
(7r i) at least a distance rj/2 on TI from the ,/-normal points on .7 g b.
Class II shall contain the remaining points (tt) on J g b.
If (tt) belongs to Class 1 and (ttj) is its final image under 1)', there will be at
least one elementary extremal of g(w i) with a ./-length less than
M {tt j) — Ci,
whore is the maximum of the ./-lengths of the elementary extremals of
g(rri) and e\ is a positive constant independent of the point (7r) in Class I. We
then have
«/0n) S (p + l)Af (tt i) — ei.
But from the definition of /)',
MUi)(p + 1) ^ 7W,
so that
J(tti) g J (tt ) — Cl.
Now’ J (tt\) will not be increased under I)", and we see that J{tt) is accordingly
decreased by at least ex under D”l)' if (tt) initially belongs to Class I.
If (t) belongs to Class II, but is not a critical point, J(tt) is decreased under
D" as follows from (3.7).
The lemma is thereby proved.
We need to represent the ensemble of points (7 r) with p + 2 vertices as a com¬
plex. With that in view recall that the pairs of end points (A1, A2) are the
images of points on the terminal manifold Z, while the intermediate vertices
lie on J?. The totality of points (77-) can accordingly be represented by the
product complex
Z X np.
With this understood we can prove the following theorem.
204 THE BOUNDARY PROBLEM IN THE LARGE [ VII ]
Theorem 3.1. If b is an ordinary value of the connectivities of the domain
JM < b are finite.
First observe that the boundary of the domain J(tt) < b consists of points at
which
or at which
JM = 6,
MM = p,
where M(ir) is the maximum /-length of the elementary extremals of g( tt). We
shall prove that the connectivities of the domain J(t) < b are finite by showing
that Dp deforms the domain J(t) ^ b on itself onto a complex on its interior.
In particular under D' any point (7 r0) will be deformed into a point (7r) for
which
MM g -L- < p.
V + 1
Moreover under D" points at which M(w) < p are deformed through such
points. Hence under Dp all points on the domain J(tt) ^ b are carried into
points for which M(tt) < p.
On the other hand points (t) at which J(tt) — b are ordinary points by
hypothesis, and by virtue of Lemma 3.1 are carried into points (w') at which
J(tt') < J( tt) - b.
In sum Dp deforms the domain J(7r) < b on itself into a subdomain H at a
positive distance from the boundary of the domain J(ir) < b. But if the
product complex Z X Rv is sufficiently finely divided, a subcomplex C of its cells
can be chosen so as to include the points of H and to be included on the domain
J(t) < b. By virtue of the deformation Dv any cycle on J(tt) < b is homologous
on /(tt) < b to a cycle on C. Since all cycles on C are homologous on C to a
finite set of such cycles, the same is true of cycles on J(w) < b and the theorem is
proved.
We continue with the following theorem.
Theorem 3.2. If a and b, a < b, are two ordinary values of J between which
there are no critical values of J, the connectivities of the domains J(tt) < a and
J(tt) < b will be equal .
The proof of the preceding lemma makes it clear that under Dp each point (tt)
on the domain
a ^ J (w) 5s b
is carried into a point {*') such that
JM) < JM - d,
[4]
RESTRICTED DOMAINS ON n
205
where d is a positive constant independent of (tt). Hence if Dp is repeated a
number of times n, such that
dn > b — a,
the domain J(ir) <6 will be deformed on itself onto the domain J(ir) < a.
Hence any cycle on J(ir)<b is homologous on J(tt)< b to a cycle on J( n) < ay
so that the connectivities of J(tt) < b are at most those of J(tt) < a. But
any set H of ./-cycles on J(7r) < a between which there is no proper homology on
JM < a, will likewise admit no proper homology on J( tt) < b. For otherwise a
use of the product deformation Z)£ would lead to an homology between these
same cycles H on J(tt ) < a. Thus a maximal set of ./-cycles independent on
JM < a is a maximal set of ./-cycles independent on J(tt) < b. The number of
./-cycles in such a maximal set is the common ./-connectivity of the domains
J(tt) < a and J(tt) < b.
Restricted domains on II
4. Let zj be a ./-chain on the functional domain 0. Let a, be a ./-cell of z;.
The closure u, of a, can be represented as the continuous image on R of the closed
functional simplex a} X t\ of §1. Let P be a point on The image on a, of
the product P X ti on a; X t\ has been termed the curve on a, determined by P
on a,-. If the curves on a, determined by points P on a, are restricted curves
on which the J-length of each curve from its initial point to the image of a point
Q on aj X t\ varies continuously with Q, ay will be termed a restricted j-cell. If
Zj is a sum of the closures of restricted ./-cells, Zj will be called a restricted j-chain.
Employing restricted chains and cycles only, one can now formally define the
connectivities of SI as before. We term these connectivities the restricted con¬
nectivities of SI. We shall prove the following theorem.
Theorem 4.1. The restricted connectivities Ri of the functional domain SI equal
the corresponding unrestricted connectivities Pi of SI.
Let kj be an unrestricted chain on SI. Let y be any one of the curves of kj .
Let p be a positive integer and let y be divided into p + 1 segments of equal
variation of t. Let (7r) denote the point determined by the successive ends
of these segments of 7, and let h be any one of these segments. If p is sufficiently
large (and we suppose it is), each point of h can be joined to the initial point of h
by an elementary extremal ^ Moreover we can suppose that p is chosen
independently of the curve 7 of kj under consideration. With p so chosen we
shall now define certain deformations.
The deformation 5'. We shall deform the preceding curve 7 into g( t). Let r
represent the time during this deformation with O^rgl. For each such value
of r we suppose h is divided into two segments X and X' in the ratio of r to 1 — r
with respect to the variation of ton h. For each value of r we replace the second
of these segments of h by itself, while we replace the first by the elementary
extremal ^ which joins its end points. We make a point on X which divides X
206
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
in a given ratio with respect to t correspond to the point on n which divides m
in the same ratio writh respect to the variation of J, assigning to this point on n
the same value of t as its correspondent on X bears. We denote this deformation
by b'.
The deformation b" . Let g be an arbitrary restricted curve. The deformation
<$" will not change g except in parameterization. To define <5" we let the point t
on g move along g to the point on g which divides g with respect to /-length in
the same ratio that t initially divided the interval (0, 1), moving at a constant
/-rate along g equal to the “/-length on g” of the arc of g to be traversed. In the
resulting parameterization the parameter t again runs from 0 to 1, but is now
proportional to the /-length of the arc of g preceding the point t. Such a param¬
eterization will be termed a J -parameterization.
The deformation 8,,. We define <5;, as the product
bp = b"b'
following 5' by the deformation <5".
Let an be an h- cell of A:, given as the image on R of a functional simplex ah X G*
Let y be the curve on dfl which is determined by the point P on a/t, and let yT
be the curve which replaces 7 at the time r under the deformation bp. Let tj
denote the interval 0 ^ r ^ 1. If the point t on yr be regarded as the image of
the point (P, tf r) on the product
oth X ti X r 1,
we see that bp defines a deformation of ah on 12 in the sense of §1 .
We now come to the proof of the theorem.
We shall denote the restricted ./-chain into which an arbitrary ./-chain kj on 12
is deformed under bp by r(k}). Observe that when
kj — > Ar;_i (on 12)
we have
(4.1) r(kj) -> r(fc,-_i).
If A;, is a j- cycle on 12 we see that r{kj) will be a restricted ./-cycle, homologous to
kj on 12.
It follows that Pi ^ Hi and that Ri must be infinite with Pt.
To show that P, = Pt-, we have merely to showr that a restricted j-cycle Zj on 12
which bounds an unrestricted chain z]+l on 12, necessarily bounds a restricted
chain on 12. We are supposing then that
Zj + L-tzj (on 12).
If the integer p is sufficiently large, the deformation bp is applicable to 27+1 and
we have
(4.2)
r(zj+i) -*• rfa)
[4]
RESTRICTED DOMAINS ON il
207
Let ! be the chain through which the cycle z3 is deformed under bp. We
have (always mod 2),
(4.3) wi+i -» Zj + r(z,),
and hence from (4.2) and (4.3),
(4.4) r(zj}l) + wJ + 1 — ► 2y.
Moreover the left member of (4.4) is a restricted chain since z3 is a restricted
cycle. Hence c; bounds a restricted chain if it bounds at all on 12.
We conclude that IJt — Ri and the theorem is proved.
The restricted domain ilb. The set of restricted curves on il on which J < b
will be denoted by ih- Concerning il b w e now prove the following theorem.
Theorem 4.2. The restricted connectivities R\ of Ub equal the connectivities R *
of the domain J (n) < 6. The connectivities R { are thus independent of the number
of vertices p + 2 of their points (tt), provided only (p + 1 )p > b.
To prove this theorem we shall begin by showing that any chain c, of points
(ir) on J (tt ) < b leads to a chain 12(c;) of restricted curves on 12 &.
The chain 1 2(c,). Let fry be aj-cell, the image on the domain n of an auxiliary
j-simplex «y. Let (tt) be a point on fry and p its image on a,. Suppose that
g(ir) has a ./-parameterization, with parameter t. We make the point t on g(ir)
correspond to the point (p, t) on ay X ti. We have thus defined a continuous
image of ay X t\ on R , or if we please, a closed j-cell on 12 derived from fry. We
denote this closed j-cell on 12 by 1 2(fr;). With the chain c, on II we now associate
the sum of the closed j-cells 12(6,) on 12 “derived” from the respective j-cells of cy.
We denote this chain on 12 by 12(c,). We shall refer to the integer p giving the
number of intermediate vertices of the curves g( t) making up 12(c;) as the index
of 12(c;). We see that t2(c;) will be a cycle on ilb if and only if r, is a cycle on
the domain of the points (x).
I say that R) g R". For any restricted j-cycle on 12 b will be deformed under
8P into a cycle 12(c;) of index p, derivable from a cycle c} of points (w). But the
cycle Cj will satisfy a relation (mi = 1, or 0)
Cj- i-i * Cj m*y* ii 1, ■ > R y)
in w7hich yi is the tth cycle of a set of R ” j- cycles forming a maximal set of j- cycles
independent on the domain J{ t) < 6, and cy+J is a (j + l)-chain on J(t) < b.
We then have the relation
12(cy+1) — ► il(Cj) + mi12(71') (on 12 b),
or
il(cj) ~ miil(yi) (on 12 b).
Hence R '■ is at most the number of cycles 12(<yi), that is, at most R* .
To conclude that R '• = R* one has merely to prove that a cycle 12(cy) of index
208
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
p, “derivable” from a cycle c} of points (tt) on the domain J(tt) < b, bounds a
restricted chain &,.fi on 12 & only if c, bounds a chain on J(ir) < b. To that end
we shall now show how a chain z, of restricted curves on Qb leads to a chain i r(z7)
of points (7 r).
The chain 7 t(z2). Let h be any restricted curve for which J < b. The point
(w) whose vertices divide h into p + 1 segments of equal variation of / will be de¬
noted by 7 r(h).
Let bj be a restricted j-cell on 12 bl the image on R of a functional simplex
ctj X U. Let h be a restricted curve of 6; determined by an arbitrary point p of
aj. The point 7 r(h) will now be regarded as the image of the point p on a,.
We thus have a continuous image of a, among points (7r), or if we please a closed
j-cell 7 r(bj) on the domain /(tt) < 6, derived from the ,7-cell b} on 12&. More
generally a chain z2 on Q. b shall be regarded as determining that chain 7r(z?) on the
domain /(tt) < b which is the sum of the closed j-cells 7r (6,) derived from the
respective ./-cells 5, of z2.
Suppose that a cycle O(cy) on 12 b of index p bounds a restricted chain fcy+j on
12 b. On the domain J( t) < b we then have
(4.5) j -j- 1) 7t(12(c,)).
Let (?r) be any point on the cycle c} and g( n) the corresponding J-parameterized
broken extremal. The point T(g(ir)) consists of vertices which lie on g( 7r). The
point (7 r) can be deformed into the point Tr(g(ir)) by moving its vertices along
g( 7r) to the corresponding vertices of 7r(gf(7r)), moving each vertex at a J-rate
equal to the /-length on g( t) of the arc of g(ir) to be traversed. The cycle e3 will
thereby be deformed through a chain + 1 on ,/( w) < b into the cycle 7r(12(c,)).
We accordingly have the relation
Wi+I — ’ c, + ir(0(c,)),
and upon using (4.5) we find that
7r(/c;+l) + Wi+1 -> cr
Thus the cycle c7 of points ( ir ) bounds on J(t) < b if 12(c,) bounds on 12 5. It
follows that R] = RJ f and the theorem is proved.
The preceding theorem taken with Theorem 3.2 gives us the following.
Theorem 4.3. If a and 6, a < 6, are any two ordinary values of J between which
there are no critical values of J, the restricted connectivities of the functional domains
J < b and J < a are equal.
The /-distance between restricted curves
5. An unordered pair of points on our Riemannian space R will be said to
possess a /-distance equal to the inferior limit of the /-lengths of restricted curves
joining the two points. We see that the /-distance between two points of R
varies continuously with the points.
[ 5 ] THE /-DISTANCE BETWEEN RESTRICTED CURVES 209
We shall now define the J -distance between any two sensed curve segments gx
and g2 of class Dl.
To that end regard points on g i and 02 as corresponding if they divide g i and
02 in the same ratio with respect to the /-lengths of their arcs. We now define
the /-distance d(gu 02) between g , and 02 as the maximum of the /-distances
between corresponding points of gx and 02 plus the absolute value of the differ¬
ence between the /-lengths of gx and 02. Cf. Fr6chet [1], We see that
d(g i, 02) = Z(02, 01).
This definition has the advantage that it is invariant of coordinate systems and
that under it
I J 01 I
is arbitrarily small if d(gu g2) is sufficiently small. Moreover it corresponds, as
will appear shortly, to vital needs of our developments, particularly in connec¬
tion with the deformations of restricted chains, where the notion of “uniform
/-continuity” is introduced.
If 03 is a third sensed curve of class Dl we have the triangle relation
d(g i, 0s) ^ d(gu 02) + d(g2t 03).
With the aid of this relation we see that if g2 is sufficiently near g%} that is, if
d(02, 03) is sufficiently small, d(g\9 gz) will differ arbitrarily little from d(gXy g2).
Let 0i be any restricted curve. By a neighborhood of 0i on 12 will be meant a
set of restricted curves which includes all restricted curves within some small
positive /-distance e of gx. Let A be a set of restricted curves of 12. The curve
0i will be called a limit curve of curves of A if there is a curve of A in every neigh¬
borhood of 0,. The boundary of A is the set of restricted curves which are limit
curves of curves of A as well as of 12 — A. Open, closed, and compact sets on 12
are now defined in the usual way. Particular examples of closed and compact
sets A are restricted chains and critical sets of extremals.
If A and B are any two sets of restricted curves, d(A, B) will be defined as the
inferior limit of the /-distances between curves of A and B. If A and B are
compact, d(A} B ) will be taken on by at least one pair of curves in A and B
respectively. The distance d(g j, A) varies continuously with gl9 that is, it
changes arbitrarily little if gx is replaced by a restricted curve sufficiently near 0i.
We shall continue with the following lemma.
Lemma 5.1. If a is a curve of class Dl consisting of an arc a ' followed by an arc
a", and b is a similar arc of class D 1 consisting of an arc b' followed by an arc b” ,
and if
d(a\ b') < e, d(a\ b") < e,
then
d(a, b) < 4e.
210
THE BOUNDARY PROBLEM IN THE LARGE
[ VII J
Let us denote the ./-lengths of each of the preceding arcs by the letter that
designates the arc.
Under the hypotheses of the theorem, | a — b | < 2c. Let A ' be a point on a'
at a ./-distance ta' on a ' from the initial point on a\ 0 g t g 1. The point A '
on a' will “correspond” to the point B' on b' for which the /-distance from the
initial point of b' is tb' . Regarded as a point on a, A ' will correspond to the
point B on b whose /-distance on b from the initial point of b will be
ta'
F+a7'
(bf + b").
The /-length of the arc of b between B and B' is then seen to be
ta
, (b' + b")
[a' + o")
Upon setting
V = V
a ,
lb'
b" - a*
the distance r takes the form
t(aY - aW)
' a' -+ «"
< / ( — I I + a" 1 rL ^
= V ~a' + n" )
and this is at most c, since | r\" | and ! 77' j are at most c. But tl e /-distance
| A 'B | is at most the sum of the /-distances | A'B' | and i B'B J and is accord¬
ingly at most 2e. Similarly the /-distance between a point A" on a, which is
given as a point on a", and the corresponding point on b is less than 2c. The
lemma now follows from the definition of d(a, b).
Let F be any deformation which deforms each restricted curve (h in the ordi¬
nary sense through a 1-parameter family of continuous curves depending con¬
tinuously on the curve parameter t and the time r, 0 g t ^ 1. We shall say
that F is uniformly J -’Continuous over a set B of restricted curves, if correspond¬
ing to a positive constant e, there exists a positive constant 77, so small that any
two restricted curves whatsoever of B within a distance 77 of each other when
r = 0, remain within a distance e of each other at each subsequent moment r of
the deformation F.
In terms of the previously defined deformations 5' and 5" we now introduce
the deformation
a;, = h"b'b\
and prove the following lemma.
Lemma 5.2. If b < (p + 1 )p, the deformation A'p is uniformly J -continuous
over the set of restricted curves Qb on which J < b.
[51
THE /-DISTANCE BETWEEN RESTRICTED CURVES
211
Recall that 5" alters a restricted curve merely in parameterization, leaving
the curve at a zero /-distance from itself. Under 8" the final image of a curve
acquires a “/-parameterization”.
Let g be a restricted curve on with a /-parameterization. The definition
of the deformation <$' involves breaking g into p + 1 successive arcs of equal
variation of t, here of equal variation of /. At the time t, 0 g r ^ 1, the
gth one of these arcs is divided into two arcs X and X', here in the /-ratio of
r to 1 — r, and the first of these arcs X is replaced by an elementary extremal y
joining its end points. Let gx be a second restricted curve on ilb, possessing a
/-parameterization, and let Xu X J , and yx be related to gx at the time r in the
definition of S' as X, X', and y are related to g .
Suppose that
(5.1) d(g , qx) < v
where rj is a positive constant. If the points which divide g and gx in the same
/-ratio correspond, the respective subsegments X' and \[ of g and gx will in
particular correspond. It follows that
(5.2) d(X', Xj) < v.
Let e be an arbitrarily small positive constant. If (5.1) holds, the respective
end points of y and y\ are within a /-distance rf of each other, and hence if rj is
sufficiently small,
(5.3) d(y, m) < e
uniformly for all curves g and gx on
The arc y followed by the arc X' forms a curve which we denote by y + X'.
The arcs yx and \[ similarly form a curve yx + \[. If we suppose 77 < e, as we
very well can, it follows from (5.2), (5.3), and Lemma 5.1 that
(5.4) d(y + X', yx + Xx) < 4e.
That is, the /-distance between the curves which under 5' replace the #th seg¬
ments of g and gx at the time r is at most 4e.
Let g* and /’f denote the curves which under S' replace g and gx respectively
at the time r. Upon regarding <7* and g* as the sum of p + 1 segments such as
appear in (5.4), and applying Lemma 5.1 p times, we see that
d(g*> <7?) < 4p(4e).
Thus the /-distance between the deforms of g and gx under S' remains uniformly
small if the /-distance between g and gx is initially sufficiently small.
Following the deformation 8' 8" by the deformation 6" will further change the
restricted curves only in parametrization. The proof of the lemma is now
complete.
212
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
Cycles on 17 neighboring a critical set co
6. By a critical set of extremals co we mean a connected set of critical extremals
on which / equals a constant c and which are at a positive /-distance, in the
sense of the preceding section, from other critical extremals. If co contains all
of the critical extremals on which / = c, co is called complete. In the present
section a? may or may not be complete.
By a neighborhood N of co will be meant an open set of restricted curves which
includes all restricted curves within a small positive /-distance c of w. We
admit only such neighborhoods of to as consist of curves whose /-distances from
other critical sets of extremals is bounded away from zero. We also suppose
that the curves of N satisfy the condition
(6.1) a < J < b,
where a and b are two constants which are not critical values of / and between
which c is the sole critical value.
By a /-normal curve of index p we mean a /-parameterized curve g(ir) deter¬
mined by a /- normal point (7 r) of p + 2 vertices. With this understood we state
an analogue of Theorem 3.1, Ch. VI.
Theorem 6.1. There exists a deformation 6p(t) of restricted curves which is
defined and continuous for restricted curves sufficiently near to and for t on the interval
0 ^ t < 5, and which has the following properties .
It deforms extremals of co on themselves , and for t S 4 replaces each curve by a
J -normal curve of index p. Any sufficiently small neighborhood N of co is thereby
deformed into a neighborhood Nt, the superior limit of the distances of whose curves
from co approaches zero as t approaches 5. Restricted curves below c are deformed
through such curves.
The proof of Theorem 6.1 is more difficult than the proof of Theorem 3.1,
Ch. VI. The method of proof will be first to deform each restricted curve
neighboring co into a curve g(it) for which (it) is /- normal. This deformation will
then be followed by a deformation of /-normal points (tt), and of the correspond¬
ing curves 0(71-). We shall devote the next two sections to the development of
these ideas.
Let N* be a fixed neighborhood of co whose closure is interior to the domain on
which 6p(t) is defined.
We state the following corollary of the theorem.
Corollary. Corresponding to any neighborhood X of co on N* let M(X) be a
neighborhood of co so small that M(X) is deformed under 0p(t) only on X. Each
restricted k-cycle on M(X) ( below c) will then be homologous on X ( below c) to a
k-cycle ( below c) on an arbitrarily small neighborhood N of co. If zk ~ 0 on N*
(below c), and is sufficiently near co, then zk~0onN ( below c ).
[7]
THE SPACE 2 OF /-NORMAL POINTS
213
The phrase (below c ) can be read throughout or omitted at pleasure.
The space 2 of /-normal points
7. We continue with the critical set <*>. On extremals of a >, / = c. We are
supposing that the number p + 2 of vertices in points (t) is so large that
(p + 1) p > c,
where p is the superior limit we have set for the /-lengths of elementary extrem¬
als. Let cr* be the set of /-normal points of p + 2 vertices determined by a>.
We shall prove the following theorem.
Theorem 7.1. The J-normal points (7r) of p + 2 vertices in a sufficiently small
neighborhood of a* make up a regular , analytic , Riemannian subspace 2 of the
(p + 2 )-fold product of R by itself .
Let (ttq) be any point of v*. We shall first prove that the /-normal points
neighboring (7r0) admit a regular analytic representation of their vertices.
Set g( 7T0) = g. Let s be the arc length along g . Let
a0 < a1 < • • • < a* < ap+1
be the values of s on g at the vertices of (ir0). Neighboring the point s = aq
on g let R be referred to local coordinates
(7.1) (xq, y\} • ■ • , yl) (w = m - 1)
in such a manner that 2/?= ••• = 2/n==0 and 6* = xq along g neighboring s = aq.
Let (7r) be any point neighboring (? r0). Let the /-length J q of the gth elemen¬
tary extremal of q(tt) be regarded as a function of the coordinates of the #th and
( q — l)th vertices of (7 r) in (7.1). In the special cases of /0 and /p+ 1 respectively
we understand that the coordinates of A1 and A2 are replaced in J0 and Jp+ 1 by
the functions of the end parameters (a) which give admissible end points A1, A2.
The conditions that (w) be a /-normal point are that
Jo — / 1 — 0,
(7.2) .
/ p~~ 1 Jp — 0,
where the variables involved are the variables (7.1) for q = 1, • • • , p, and the
end parameters ah .
The parameters ( u ). We shall show that the equations (7.2) can be solved
for the variables
(7.3)'
as analytic functions of the remaining variables
(7.3)"
(u) = [y\, a*]
214
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
To continue we consider the jacobian D of the left members of (7.2) with
respect to the variables (7.3)'. We evaluate this jacobian at (7r0).
Let /(s) be the value of J taken alone g from s — aP to the variable end point s .
We set
/'(«") = S'„ (? = !,•••, P)-
We make use of the fact that we can set the variables (u) equal to their final
values before computing the partial derivatives of the left members of (7.2)
with respect to the variables (7.3)'. If we consider the typical case where there
are five variables in (7.3)', we find that at (w0) the jacobian /> is given bv the
equation
2/;
-ft
0
0
0
-/;
2/;
J
0
0
I) =
0
--/ 2
2J
r'
-/;
0
0
0
"J
2/.;
f b
0
0
0
-/;
Since/' is never zero,
T) vanishes
, if at
all, w
it h the
del*
mth inant
2 —
1
0
0
0
-1
2
-1
0
0
0 - 1 2 - 1 0
0 0 -1 2 -1
I 0 0 0 -1 2
One can easily show7 that determinants of this form are never zero.
We can accordingly solve the conditions (7.2) for the coordinates (7.3)' as
analytic functions of the variables ( u ).
Now' the set of coordinates
[*q, yq ] 0/ = o, • • • , p + i)
make up a local coordinate system for the (p + 2)-fold product of R by
itself. One sees that the conditions (7.2) together with the end conditions on
A1, A2 define a regular analytic subspace 2 of Rp+2, at least neighboring (ir0),
and that t he preceding parameters (u) may be regarded as local coordinates on 2
neighboring (tt0).
Neighboring (t0) we can assign Rp+2 a local metric, defining this metric by a
differential form ds2 which is the sum of the forms defining the metrics of R
neighboring the respective vertices of (7 r0). One can then assign 2 a submetric,
making arc length on 2 agree with arc length on Rp+2, taking the parameters ( u )
as local coordinates of 2.
[7]
THE SPACE £ OF ./-NORMAL POINTS
215
The t heorem is thereby proved.
We now come to the following theorem.
Theorem 7.2. The value of J{ w) on the subspace 2 of J-pormal points (i r)
sufficiently near a* is an analytic function of the local coordinates (u) of 2 and
possesses no critical points other than points of the set a*.
Let v be any one of the coordinates (u). At a critical point ( u ) of J (ir) on 2
we have
dJ o
3v
+ +
dv
+ dJ~i‘ = o
dr
Since the equations (7.2) become identities in the variables (u) on 2, on 2 we
have
dJ o dJ i
dv dv
dJ _ d J j, _
dv dv
Combining these conditions on a critical point we see that
(7.4) ~" = ° (<? = 0, 1, ••• , V)
at each critical point (v).
In terms of the local coordinates (x\ yl) = (x} y) of R neighboring the vertex
Pl of (7 to), the conditions (7.4) include in particular the conditions
(7.5)
dJ 0 __ dJ 1
dy\ ~ dy\
(i == 1, n).
1 say that the conditions (7.5) imply that the broken extremal g corresponding to
the critical point (u) has no corner at the vertex Pl of g.
To prove this statement observe that on 2 near (7r0) the local coordinate xl
of the point (x\ y}) is an analytic function X(u) of the local coordinates (u) of 2,
and in particular an analytic function of the local coordinates y\ . If the
integrand of J be put in the non-pa rame trie form f(x, y , p) in the local system
(.r, y) = (A if) we see that, for h = 1, • • • , n,
(7.0)'
(7.6)"
[(/- +A.]
dJ 0
dy\
dJj
= 0,
0,
216
THE BOUNDARY PROBLEM IN THE LARGE
l VII]
where the arguments (xf y, p) corresponding to the upper limit are the co¬
ordinates Or1, yl) and slopes pi = X< at the final end point of the first elementary
extremal of g, while those corresponding to the lower limit are the same coordi¬
nates (s1, yl) with the slopes pi = yi ; at the initial end point of the second ele¬
mentary extremal of g. We set
(Xi - n) \u- r»fvh) ~ + 4 1 <X> = "(X, /*)•
L by \ J ca*>
We consider //(X, y) as a function of the variables (X) for (X) near (y), and ex¬
pand ff (X, y) as a power series in the differences X» — ytJ holding (x\ yl) and (y)
fast. WTe see that
H(y, y) = 0 HXt(y, y) = 0 (i = 1, • • • , n),
and that
HXhXk(y, y) « 2fPhn(x\ y\ y) + 2^, y\ y)
where rj(xlt yl , y) tends to zero as (y) tends to (0). Accordingly for h} k — 1,
• • * , n,
(7.7) //(X, y) = l/pAPjt(xl, y1, y) + 77J (Xh - yn) (X* ~ m*) + * * ■ ,
where the terms omitted are of higher order than the second in X; — yy
We now see that (7.6) cannot hold for (X) and (y) sufficiently near (0) unless
(X) = (y). For if (7.6) held for sets (X) and (y), H(\, y) would vanish for these
sets. From (7.7) it would follow that (X) — ( y ) if the broken extremal g lies
sufficiently near <7(71*0). Hence g has no corner at P1, and the statement in italics
is proved.
Similarly the conditions of the form (7.4) corresponding to the remaining
vertices P% of g imply that g has no corners at these vertices. Finally the
conditions (7.4) of the form
dJ 0 __ dJ p
dah dah
(A = 1, • • • , r),
in which the variables (a) are the parameters in the end conditions, imply that g
satisfies the transversality conditions. Thus g is a critical extremal, and the
theorem is proved.
Theorem 6.1
8. In this section we shall prove Theorem 6.1 and deduce certain consequences
therefrom. We begin with the following lemma.
Lemma 8.1. There exists a deformation Ep(t)} 0 ^ t ££ 4, of the restricted curves
neighboring the critical set a> which carries restricted curves into J -normal curves , and
leaves J-normal curves invariant. Moreover Ep(t) is uniformly J -continuous over a
[8]
THEOREM 6.1
217
sufficiently small neighborhood of u>. It deforms each extremal of co on itself into the
corresponding J -parameterized extremal.
To prove this lemma we first apply the deformation Ap of §5 to the restricted
curves neighboring co. Each extremal g(ir) on co will thereby be deformed on
itself into the corresponding J-normal extremal. Since A' deforms restricted
curves for which J < b in a manner that is uniformly ./-continuous, it follows
that all restricted curves sufficiently near co wrill be deformed into broken ex¬
tremals g(ir) for which (7r) lies within a prescribed positive /-distance e of the
./-normal points a* determined by co. If e is sufficiently small, on each such
curve g( x) there wrill exist a unique set of p successive points which together with
the end points of g( x) are the vertices of a J-normal point (t r'). We term (x')
the J-normal image of (x). If the above constant e is sufficiently small, we see
that the J-normal image (x') of (x) will vary continuously with (tt).
The deformation Ap. We deform g{ x) into g(ir') letting the vertices of a
variable point (x t) move along g(ir) from tlie vertices of ( x ) to the corresponding
vertices of (i r'), each vertex of (x<) moving at a J-rate equal to the J- length
on g( x) of the arc of gin) to be traversed. We assign ,7-parameterizations to
each curve thereby replacing g{ x), and denote the resulting deformation by
Av. In Ap the time t varies on the interval 0 ^ t ^ 1.
The deformation A'p} defined as the product of deformations 5", <$', and <$", in
each of which the time runs from 0 to 1 inclusive, may itself be regarded as a
deformation in which 0 g t ^ 3.
The deformation Ep(t ). With this understood we nowT set
EP(t) = A"PA'P 4),
understanding that the deformation A' is followed by the deformation Ap.
Under Ep(t) we can suppose that Ap occupies the time interval 0 ^ ^ 3, and
that Ap occupies the time interval 3 g ^ 4, so that the time interval for
Ep(t) becomes 0 ^ t g 4. The symbol Ep(t) stands for the deformation up to
the time t. The deformation Ep(t) is applicable to all restricted curves on a
sufficiently small neighborhood of co.
That Ap is uniformly ./-continuous over any domain J < b of restricted curves
has already been est ablished, provided alwrays that {p + 1 )p > b. A sufficiently
small neighborhood of co will be such a domain. The deformation Ap is likew ise
uniformly J-continuous by virtue of Lemma 5.1, if the /-normal images (x')
of the points (x) involved depend upon (x) in a uniformly continuous manner.
But this dependence of (x') upon (x) will clearly be uniformly continuous if the
initial neighborhood of co is sufficiently small.
The remaining affirmations of the lemma require no further substantiation
and the proof is complete.
In the preceding section we have seen that the set of all /- normal points (x)
neighboring <r* make up an analytic Riemannian space 2 and that /(x) is an
analytic function of the local coordinates ( u ) of 2, with the set <r* of /-normal
218
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
points as its critical points. With this critical set a* on 2 we now associate a
neighborhood function v? of the point on 2 neighboring cr*, exactly as in §4,
Oh. VI. Let r be a positive constant so small that the points on 2 which are
connected to a* and at which
(8.1) ^ r
form a closed domain at each point of which <p enjoys the properties of a neigh¬
borhood function.
The radial deformation RP{t), 0 ^ t < 1. With the aid of the preceding
function <p we introduce radial trajectories on 2 as in §5, Oh. VI. It will be con¬
venient to say that a /-normal curve g(ir) lies on a domain ip = k (a constant),
if the point (? r) lies on the domain <p — A\ We shall now define a deformation
Rp(t) of ./-normal curves on the domain ip ^ r. Under Rp(t) the time t shall
vary on the interval 0 ^ < I. We first deform the ./-normal points (x) on
ip <£ r as follows. Let 6 be a constant such that
o g e < i.
Each / -normal point (x) at which
^ — r — dr
shall remain fixed under Rp(t) until t reaches 0, and shall thereafter be replaced
at the time t by the point (x,) on the radial trajectory through P at which
<p = r — tr. The ./-normal curve g(w) shall likewise remain fixed until t
reaches 6} and shall thereafter be replaced at the time t by fir(x,).
The deformation 6 ,,((), 0 :§ t < 5. Under the deformation Ep(t), with its time
interval 0 fg t :g 4, a restricted curve sufficiently near u> will be deformed into a
/-parameterized /-normal curve 7 on the domain <p g r. To such a curve 7
the deformation Rp(t), 0 ^ t < 1, is applicable. It is therefore legitimate to
introduce a deformation 6p{t) such that
dp(t) = Ep(t) (0^^4)
and to continue this deformation so that in so far as the curves which are ob¬
tained when t = 4 are concerned,
6p(t) = Rp(t - 4) (4 g t < 5),
thus defining on the time interval 0 g t < 5. It follows from this defini¬
tion that 6p(t) has the properties ascribed to it in Theorem 6.1.
The proof of Theorem 6. 1 is now complete.
Recall that N* has been chosen as a neighborhood of a? which is so small that
its closure is interior to the domain of definition of 0p(t). Corresponding to any
neighborhood I of w such that X C N*, we let M(X) be a neighborhood of w
so small that M(X) is deformed under 6p(t) only on X (0 ^ t < 5).
[8]
THEOREM 6.1
219
A n ordered pair of neighborhoods VW of co will now be termed admissible if they
satisfy the condition's
VCZM(N*)y W C M(V).
Spannable and critical ^-cycles corr VW are now formally defined as in Ch.
VI with the present interpretation of the terms involved. These cycles of
restricted curves will be spoken of as belonging to the critical set The phrase
corr VW will be omitted in cases where it is immaterial which pair of admissible
neighborhoods VW is used. We continue with the following analogue of
Theorem 3.2, Ch. VI.
Theorem 8.1. Corresponding respectively to any two choices VW and VW* of
admissible pairs of neighborhoods of w there exist common maximal sets of spannable
or critical k -cycles of restricted curves on any arbitrarily small neighborhood of u
Theorem 8.1 follows readily from Theorem (11.
Let e be a positive constant less than the constant r of (8.1). A fc-cycle of
./-normal curves on ip = e below c, independent on this domain, but bounding on
<p S c will be termed a spannable k -cycle of /-normal curves corr <p g e. A
A -cycle of /-normal curves on <p g c independent on this domain of A-cycles on
<p ^ e below c will be called a critical k- cycle of /-normal curves corr <p ^ e.
Maximal sets of spannable or critical fc-cycles of /-normal curves corr <p g e
exist according to the t heory of neighborhood functions ip of Ch. VI.
The analogue of Theorem 5.2 of Ch. VI can now be stated as follows.
Theorem 8.2. Maximal sets of restricted critical and spannable k-cycles corr
I r IT can be taken respectively as maximal sets of critical and spannable k-cycles of
J -normal curves corr <p ^ e, provided e is a sufficiently small positive constant.
Corresponding to the given neighborhood W we choose the constant e so that
e < r and so that the domain of /-normal curves on ^ e is on W . For this
choice of e the theorem holds. We shall give the proof for the case of spannable
A-cycles. The proof for critical A-cycles is similar.
Let (u)k be a maximal set of spannable A-cycles of /-normal curves corr ip ^ e.
We shall prove that (u)k is a maximal set of spannable A-cycles corr V W .
To that end let zk be a spannable A-cycle corr VW. By virtue of the relation
of V to Wj and Theorem 6.1, we have
Zk ~ Wk (on F, below c)
where tvk is a cycle of /-normal curves on the domain ip g e. The cycle wk is
homologous on <p ^ below c} to a sum of cycles of ( u)k . Hence zk is homologous
on F, below o, to a sum of cycles of ( u)k .
Now let Uk be any proper linear combination of the cycles of the set (u)a I
say that uk ^ 0 on F below c. For if uk bounded a chain °n V below c, an
application of 6p(t) up to a time t sufficiently near t — 5 would deform wk+ 1
below c into a chain vk+1 of /-normal curves on<^ ^ e. The cycle m would at the
220
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
same time be deformed below c through a chain Uk+ 1 of /-normal curves on <p ^ e
so that
w* + 1 + Vk+I — > Uk
on the domain <p ^ e below c. Since this is contraiy to the nature of the set
{u)k we infer that Uk ^ 0 on V below c.
The critical cycles can be similarly treated. The proof of the theorem is now
complete.
Cycles on the domains J < b and J < a
9. Having analysed the restricted cycles neighboring the critical set w we shall
now examine the changes in cycles of restricted curves with respect to bounding,
as one passes from the domain J < a to the domain J < b. We are here sup¬
posing that co is the complete set of critical extremals on which / equals a critical
value c. The constants a and b are any two constants which are not critical
values of / and between which c is the only critical value.
We shall use the deformation 8p(t) of Theorem 6.1 to define a basic deformation
A p(t), 0^K5.
The deformation A p(t). We shall extend the definition of 6p(t) so that the
resulting deformation A p(t) is /-continuous over th$ restricted domain / < b
and remains identical with 6v(t) over the neighborhood N* of §6. To that end
let e be a positive constant so small that the set of restricted curves not on N*>
but at a /-distance at most e from N*, are within the domain of definition of
0p(t). Under A (t) each restricted curve at a distance say (1 — \)e from N*,
where 0 ^ \ ^ 1, shall be deformed as in $p(t) until t = A5 and held fast there¬
after. Restricted curves at a distance e or more from N* shall be held fast under
AP(0.
We shall make use of admissible pairs of neighborhoods VW of w as previously
defined. Corr VW linkable and linking cycles are formally defined as in §6, Ch.
VI. Lemmas 6. 1-6.4 of Ch. VI then hold with the interpretations of the
present chapter, the proofs remaining formally the same. One naturally replaces
points by restricted curves, and the deformations 6(t) and A (t) of Ch. VI by the
deformations 9p(t) and A p(t) respectively.
The Deformation Lemma of §6, Ch. VI, is here replaced by the following
lemma.
Deformation Lemma. Lei N be an arbitrary neighborhood of w and L the set of
restricted curves below c. There exists a J -deformation Ap of the restricted curves on
J < b which is uniformly J -continuous on J < 6, which deforms extremals of u> on
themselves, and which deforms restricted curves on J < b into rectricted curves on
N + L.
If a cycle Zk lies on a restricted domain No + L for which No is a sufficiently small
neighborhood of a> and if Zk~0 on J < b ( below c ), then Zk ~ 0 on N + L (below c).
[10]
THE EXISTENCE OF CRITICAL EXTREMALS
221
To establish this lemma we begin by applying the deformation A ' of §5 to the
restricted curves on the domain J < b. The resulting curves will be /-param¬
eterized broken extremals determined by points (7 r) with p intermediate
vertices. To these points (x) we now apply the deformation Dp of §3, and let
D'p denote the deformation of the curves g(ir) thereby generated. In D'p we
understand that a /-parameterization has been given to each curve which re¬
places g(x).
The curves g(ir) to which D'p is applied are such that /(x) < b . Such of these
curves as are not extremals of a> and for which a < /(x) will be lessened ib
/-length under D'p) as follows from Lemma 3.1. One can then prove exactly
as under the Deformation Lemma of §6, Ch. VI, that a product deformation
l)pri for which n is a sufficiently large positive integer will deform these curves
g(ir) below c onto the domain N + L. Hence the product deformation
= D'pnA'p
will /- deform the restricted curves of / < b on J <. b onto the domain N + L.
Moreover a'p is uniformly /-continuous over the restricted curves of / < 6,
as we have sem. It follow s in similar fashion that D'v is uniformly /-continuous
over the domain of curves g(n) for which /(x) < b, so that Av is uniformly
/-continuous over the restricted domain J < b.
To establish the final statement of the lemma, observe that Ap deforms a
restricted curve representing an extremal of w into the same extremal of u.
From the /-continuity of Ap it then follows that there exists a neighborhood No
of oj which is so small that restricted curves on N 0 are deformed under Ap only
on N. Suppose then that the cycle zk of the lemma bounds a chain zk+ 1 on / < b.
The deformation Ap w ill carry zk+i into a chain on N + L, deforming zk on
N + L. Hence if zk ~ 0 on / < b (below7 r), it follows that zk ~ 0 on N + L
(belowr c).
The proof of the lemma is now complete.
As in Ch. VI an invariant fc-cycle corr VW is here defined as a ft-cycle below c,
independent below c of spannable fc-cycles corr VW. Using the preceding
Deformation Lemma where we formerly used the Deformation Lemma of §6,
Ch. VI, we see that Theorem 6.1 of Ch. VI and its proof hold here in essentially
the same form as in Ch. VI, / replacing/. For the sake of reference wre repeat
the theorem in this place.
Theorem 9.1. A maximal set of restricted h-cycies independent on J < b is
afforded by maximal sets of critical , linking , and invariant k-cycles of restricted
curves corresponding to an admissible pair of neighborhoods VW of the critical set a).
The existence of critical extremals
10. Let c be a critical value of / and e a positive constant so small that c is
the only critical value on the closed interval (c — e, c + e) . Let co be the com¬
plete set of critical extremals on which / = c. Relative to the critical value c,
222
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
restricted A>cycles which are independent on J < c + e of k- cycles below c will
be termed new A>cycles. By virtue of Theorem 9.1 a maximal set of such cycles
can be obtained by combining maximal sets of critical and linking fr-cycles
corresponding to an admissible pair of neighborhoods VW of o>.
Relative to the critical value c we shall also consider maximal sets of restrict ed
(k — l)-cycles independent below e but bounding on J < c + e. We term such
cycles newly-bounding ( k — l)-cycles. It follows from Theorem 9.1 that a
maximal set of such cycles will be afforded by a maximal set of spannable
(. k — l)-cycles corr VW which are independent below c.
The number of A-cycles in a maximal set of new A-cyeles depends upon more
than the neighborhood of co, as examples would show. The same is true of the
number of cycles in a maximal set of newly-bounding (A* — l)-cycles relative to c.
It is a remarkable fact , however , that the sum mk of the number of new k-cycles
and newly-bounding (k — 1) -cycles in maximal sets relative to e depends only upon
the nature of J neighboring co.
In fact if a, ft} and y respectively denote the numbers of cycles in maximal sets
of critical A'-cvcles, linking A*-cyeles, and newly-bounding (A - l)-cyeles, we see
that
mk. = a + ft + 7-
Of these numbers a depends only on the nature of J neighboring to, while ft + 7
equals the number of spannable (k — l)-cycles in a maximal set and likewise
depends only on the nature of J neighboring to. 'Thus the statement in italics
is true.
But critical ^-cycles and spannable {k — l)-cycles are well defined not only for
complete critical sets but for critical sets in general. The definition of the type
number mk can then be consistently extended as follows.
Definition. The kth type number mk of a critical set to shall be the number of
critical k-cycles and spannable (k — 1 ) -cycles of restricted curves in maximal sets of
such cycles corresponding to neighborhoods VW of the critical set co.
A critical set co which is the sum of a finite number of component critical sets
with the same critical value, possesses a type number mk which is the sum of the
A'th type numbers of the component critical sets.
Let a and b be any two constants w hich are not critical values of J, with a < b .
Let M\ be the number of linking and critical A-cycles in maximal sets associated
with the different complete sets of critical points with critical values between a
b. Let M~k be the number of newly-bounding (k — l)-cycles in maximal sets
associated with these same complete critical sets. Let A Rk be the Al h connectivity
of the restricted domain J < b minus the fcth connectivity of the restricted domain
J < a. Let Mk be the sum of the A’th type numbers of the critical sets with
critical values between a and b. By virtue of Theorem 9.1 we have
A/4 = MX + M;+1 (A* = 0,1, ...),
(10.0)
Mu = Ml + M~u, M 0 = 0.
[10]
THE EXISTENCE OF CRITICAL EXTREMALS
223
If a is less than the absolute minimum of J, the connectivities of the domain
J < a are null by convention and ARk in (10.0) equals the A;th connectivity Rk
of the domain J < b. Upon eliminating the numbers M% from the relations
(10.0) one then finds that
(10.1) Mo — Mi + * ■ * + ( — 1 )%Mi = R0 — Ri + • • • + ( — 1)\R» + ( — 1 )%M i+i»
If r is a sufficiently large positive integer, then for k > r the numbers Mk and Rk
are null. In particular M“ + 1 will be null. From (10.1) we then obtain the
following theorem.
Theorem 10.1. The connectivities Rk of the restricted domain J < b and the
sums A Ik of the kih type numbers of the critical sets of extremals with critical values
less than b satisfy the relations
Mo £ R0,
(10.2)
Mo - Mx ^ J?0 ~ R i,
Mo— M l + ' • • + ( — 1 Y M r — R 0 — R 1 + +(~1 )rR r,
for any sufficiently large integer r.
Another particular consequence of (10.0) is that
(10.3) Mk ^ A Rk.
From (10.3) we can deduce a basic theorem on the existence of extremals which
are topologically necessary. The theorem is as follows.
Theorem 10.2. If the connectivities of the functional domain U are
Po, Pi, P*, • ■ ■ ,
the sum AT of the kth type numbers of all critical sets of extremals satisfies the relation
(10.4) Nk £P* {k = 0,1, ■■ ).
In particular if Ph is infinite , AT must be infinite.
If Pk is finite, let Qa = Pa- If Pa is infinite, let Qi be any positive integer. In
either case there will exist Qk independent A-cycles on H. As we have seen in
§4 these A:-cycles are homologous on 12 to restricted A:-cycles independent among
restricted A’-cycles. These Qk ^-cycles will lie on some restricted domain J < b,
for which b is sufficiently large. If Rk is the restricted kih connectivity of J < 6,
we will have
(10.5) Rk ^ Qk.
On the other hand if Mk is the sum of the A*th type numbers of the critical
extremals for which J <by we also have
(10.6)
AT- ^ Mk.
224
THE BOUNDARY PROBLEM IN THE LARGE
[VIII
From (10.3), (10.5), and (10.6) it follows that
Nk Qkf
from which (10.4) and the theorem follow at once.
The number
(10.7) E\ - Ni - Pi (t = 0,1, •••)
will be called the count of critical extremals of index i in excess of those topologically
necessary .
We shall investigate the limitations on these numbers Ft. We begin with the
following lemma.
Lemma 10.1. If the connectivities P o, P i, * • • , P T of il are finite , Merc trz’ZZ a
restricted domain J < b whose connectivities R{)} • • • , Fr are af F0, • • • , Fr
respectively.
If then 0 is a sufficiently large positive constant greater than bf if Mk is the kth
type number sum of critical sets on J < b} and M k the corresponding sum for critical
sets on J < 0j there will exist an integer p k between Mk and M k inclusive such that
Mo ^ Po ,
(10.8)
Mo — Mi ^ P o “ P\f
Mo ~ Mi -+■ + (-l)V $ Fo - Pi + + ( — l)rFr,
where the sign > or < is understood in the last relation according as r is even or odd m
The first statement in the lemma is a consequence of previous remarks.
To prove the second statement of the lemma we first note that
(10.9) Rk = Pk + qk (k = 0, • * , r)
where qk 0. For a sufficiently large constant 0 there must then exist a set ( h )
of qk (k = 0, • • • , r) restricted A:-cycles, independent on J < 6, but bounding
on J < 0. I say that the lemma holds for this choice of 0. We begin by proving
statement (a).
(a). The sum of the numbers of newly-bounding k-cycles in maximal sets of such
cycles relative to the critical values between b and 0 must be at least Qk (fc = 0, • • • , r).
Suppose Ci < • * • < cv are the critical values of J between b and 0. Of the
k-cycles which are newly bounding relative to ct suppose there are in a maxi¬
mal set. If qk > Pu there will be a set of at least qk — p i fc-cycles on J < cx
dependent on J < Ci upon cycles of (h) and independent on J < c2. If qk >
Pi + p2, there will similarly be a set of at least qk — pi — P2 fc-cycles on J < c2
dependent upon J < c2 upon cycles of (h) and independent on J < c$. Proceed¬
ing in this fashion we see that if
t = [qk — Pi — pt — * * — pj > 0,
[10]
THE EXISTENCE OF CRITICAL EXTREMALS
225
there will be at least t A>cycles on J < cv, dependent on J < cv upon cycles of
(h) and independent on J < &. But all cycles dependent on ( h ) are bounding
on J < £ so that t ^ 0.
Statement (a) is thereby proved.
From statement (a) as applied to (k — l)-cycles we see that
M k ^ Mk + qk~ i (k — 0, • • • , r).
We set
fik — Mk + Qk-\ (k — 0, • • • , r)y
and observe that
Mk ^ ^ Mk.
We now substitute the right members of the relations
Mk — fik — (jh~ i,
— Pk Qk,
for Mk and Rk in (10.1). Relations (10.8) are thereby obtained, and the lemma
is proved.
The following theorem is a generalization of Theorem 1.1 of Ch. VI.
Theorem 10.3 Let No, Ni, • • • be the type number sums for all critical sets of
extremals , and Let P o, P i, • • • be the connectivities of the function space il. If the
numbers Ni are finite, they satisfy the infinite set of inqualities
(10.10)
No ^ P 0,
Aro ™ N i S Po - Pi,
No - AT + N2 ^ Po - Pi + P 2,
If all of the integers Ni are finite for i < r + 1, the first r + 1 relations in (10.10)
still hold .
The first statement in the theorem is a consequence of the last. We shai!
prove the last.
Suppose that the integers
No, • • ■ , Nr
are finite. It follows from (10.4) that the connectivities
L*Q) * * * , P r
of 12 must be finite. We can then apply the preceding lemma. In applying this
lemma we can take b so large that the type numbers
wo, • • • , m
THE BOUNDARY PROBLEM IN THE LARGE
226
[VII]
of the critical sets for which J > b are all null, for otherwise some of the numbers
No, • • * , Nr would be infinite. With this choice of b the numbers
MO, * ' ‘ , Mr
in the lemma must be the numbers N0, • • • , Nr respectively. The first r + 1
relations in (10.10) then follow from (1(1.8).
We shall now prove the following theorem.
Theorem 10.4. If the connectivities P0 , P i, * * * of ft are finite, and if Ei is the
count of extremals of index i in excess of those topologically necessary , we have the
relations
Eu , + Ex-\ ^ Ei (i = 0, 1, •-■).
In particular if Ei is infinite, at least one of the two numbers Em and Ex i must
be infinite.
To prove the theorem we refer to relations (10.8). Upon combining each
inequality in (10.8) with the third preceding inequality we find that
(io.il) (M<+i - Pi+i) + (m * - 1 - p..i) ^ (m* - Pd a = o, i, •*•).
For i = 0 and i - 1 relations (10.11) reduce respectively to the second and third
relations in (10.8). If A\*+i, N {, and N m are finite, and we take the constant b
in Lemma 10.1 large enough we must have
At-fi = M»-fi> A t — pi, A t_i = Mi-],
and the theorem follows from (1 0. 1 1). *
If Nt is infinite, (10.11) still holds. In this case there must be infinitely many
critical sets whose fth type numbers are not null. The corresponding set of
critical values cannot be bounded above. If we take the constant b of Lemma
10.1 successively as the constants of a sequence becoming positively infinite,
the number m in Lemma 10.1 will become infinite with b and from (10.11) we
see that either pl+i or else Pi-i will become infinite with b. Thus the theorem
is true if Nx is infinite. If either A\-+i or AL_i is infinite, Ei+ 1 or i is infinite
respectively, and the theorem is again true.
The non-degenerate critical extremal
11. A critical extremal g is termed non-degenerate if the associated “index
form” of Ch. V, §14, is non-degenerate. We have seen in Gh. V that a necessary
and sufficient condition that g be degenerate in the case of non-tangency, is
that the associated boundary value problem in tensor form possess a char¬
acteristic root which is zero. The property of degeneracy or non-degeneracy is
an invariant property, that is, one independent of the local coordinate systems
employed.
Since no characteristic root of a given critical extremal will in general be zero,
it appears that the general case is the non-degenerate case. It is therefore fair
[11]
THE NON-DEGENERATE CRITICAL EXTREMAL
227
to say that any adequate theory of the calculus of variations in the large must
admit a definitive specialization in the non-degenerate case. We shall accord¬
ingly examine the non-degenerate case in the light of the general theory.
A first theorem of importance is the following.
Theorem 11.1. If g is a non-degenerate critical extremal , there is no connected
family, of critical extremals which contains both g and critical extremals other than g.
To prove this theorem we refer the neighborhood of g to normal coordinates
(xf y) as in Ch. V, § 1, with x the arc length along g and yi == 0 on g. We then cut
across g with intermediate n- planes, x cons* ant, so placed as to divide g into p + 1
segments of equal variation of x. We take p so large that each of these segments
is less than p in /-length. We refer these n- planes to their coordinates (y) as
parameters (0), and set up the index function J(v> 0) of §1, Ch. Ill, giving the
value of / along the broken extremal determined by (v) .
If g belonged to a connected family of critical extremals which contained
critical extremals other than g, there would be, critical extremals determined by
sets (?*) t* (0) arbitrarily near (0). But for each of these sets (v) the first partial
derivatives of J(v, 0) are zero. This is impossible, for when g is non-degenerate
the point (v) = (0) is an isolated critical point of the function J(v, 0). We
accordingly infer the truth of the theorem.
To determine the type numbers of g we return to the space of points (tt).
Let c be the /-length of g. We are supposing that the number p + 2 of vertices
in points (tt) is such that (p + l)p > c. Let a denote the set of critical points of
/ (tt ) with vertices on g , and (t*) the /-normal point of a. The set of /-normal
points neighboring (tt*) make up an analytic Riemannian manifold 2, as we have
seen in §7. On 2, J{r) has a critical point in (tt*). Let <pv be a neighborhood
function belonging to J(ir) on 2, and to the critical point (tt*). From Theorem
8.2 and the definition of the type numbers of g we have the following lemma.
Lemma 11.1. The kth type number of g is the number of critical k-cycles and
spannable ( k — 1 )-cycles of J -normal points (jr) on the domain 2 of J -normal points
(tt) belonging to the function f defined by J(tt) on 2, and to the critical point (7 r*) off
determined by g.
As previously, we denote the space of admissible points (71-) with p + 2 vertices
by II. We are supposing that (p + l)p > c. We now define a class of sub¬
manifolds of n with vertices neighboring <7.
Proper sections of n belonging to g. Let t be the arc length measured along g.
Let tl and t 2 be the values of t at the end point s of <7, and let
ti < • • • < tp
be a set of values of t between tl and P which divide g into a set of segments in
/-length less than our basic constant p. Let
Mq
(q = 1, • • * , V)
228
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
be a regular ( m — l)-manifold cutting g at the point t = tq without being tangent
to g. Points (x) whose intermediate vertices lie on the manifolds Mq, and whose
terminal vertices satisfy the terminal conditions form a regular submanifold S
of the space IT, at least neighboring the point (x) on S which determines g .
We term S a proper section of n belonging to g.
Let (7 r0) be an admissible point (x) which determines g. Suppose none of the
elementary extremals of g( x0) are null. If (x) is a point which is sufficiently
near (x0), there will be a unique point (x') on S, and a unique point (7 r") on 2
whose vertices lie respectively on g(ir). The points (x') and (tt") will be re¬
spectively termed the extremal projections of (7r) on S and 2.
We shall now prove the following lemma.
Lemma 11.2. Let S be a proper section of the space II belonging to g} and 2 the
manifold of J -normal points (x) belonging to g.
(a) . The points (7 r) on S on any sufficiently small neighborhood of <r can be
J -deformed on the corresponding broken eztrcrnals <7(71-) into their extremal projec¬
tions on 2.
(b) . Among points (tt) on S sufficiently near a, a k-cycle on S will bound on S
(below c)y if and only if its extremal projection on 2 bounds (below c ).
(c) The lemma also holds if S and 2 are interchanged.
To deform a point (x) on S into its extremal projection (tt") on 2 we deform
each vertex of (tt) along g(ir) to the corresponding vertex of (x*), moving the
given vertex at a J- rate equal to the /-length on ^(x) to be traversed. State¬
ment (a) follows readily.
Now let Zk be a fc-cycle on S. If Zk bounds a chain zk+i on S (below c)y its
extremal projection on 2 will bound the extremal projection of zk+ 1 on 2 (below
c), if Zk+i is on a sufficiently small neighborhood of cr.
It remains to prove that among points sufficiently near cr, zk bounds on S
(below c ) if its extremal projection uk on 2 bounds on 2 (below c). Suppose
then that uk bounds a chain uk+ 1 on 2 (below c). Let wk+ 1 be the deformation
chain generated by zk in its deformation into uk . If zk and uk+i are sufficiently
near <r, zk will bound the extremal projection on 2 of
wk+ 1 + uk+\.
One can interchange S and 2 in the preceding proof. Thus the lemma holds
as stated.
Let (u) be a set of parameters neighboring (u) = (uo) regularly representing S
neighboring its intersection with <r. Let F(u) denote the value of /(x) at the
point (x) determined by (u) The function F(u) is an index function correspond¬
ing to g, in the sense of §1, Ch. III. It follows that ( u ) = (w0) is a non-degener¬
ate critical point of F(u) of index k. Let <p8(u) be a neighborhood function
belonging to F and the critical point (u<>). If e is a sufficiently small positive
constant, there will exist maximal sets of spannable and critical fc-cycles on the
domain *pa < e of S, belonging to F and its critical point ( u0 ).
[11]
THE NON-DEGENERATE CRITICAL EXTREMAL
229
It follows from the preceding lemma that maximal sets of spannable and
critical fc-cycles on the domain < e' on S have extremal projections on S
which are maximal sets of spannable and critical fc-cycles on the domain tps < e
of S if ef is a sufficiently small positive constant. With this understood the
following theorem appears as a consequence of the two preceding lemmas.
Theorem 11.2. Maximal sets of spannable and critical k-cycles of restricted
curves , belonging to a non-degenerate critical extremal g, can be chosen among the
cycles of broken extremals g(ir) determined by points (t) on a proper section S of II
belonging to the extremal g.
We state the following corollary.
Corollary. The kth type number of a non-degenerate critical extremal g equals
the number of critical k-cycles and spannable (fc — 1 )-cycles in maximal sets of such
cycles belonging to the index function F defined by J(ir) on a proper section S of II,
and to the non-degenerate critical point of F determined by g.
Theorem 7.2 of Ch. VI taken with the preceding corollary gives us the follow¬
ing theorem.
Theorem 11.3. The jth type number of a non-degenerate critical extremal of
index k is <5* ( j = 0, 1, • • •)•
Theorem 9.1 then yields the following corollary.
Corollary. If a and b are two ordinary values of J between which there lies
just one critical value c taken on by just one non-degenerate critical extremal of
index fc, the only changes in the restricted connectivities as one passes from the domain
J < a to the domain J < b are that either
Case 1:
Ai?jt — 1,
or
Case 2:
Ai?jt-i = — 1.
Case 1 always occurs if k = 0. If k > 0, Case 1 or Case 2 occurs according as a
spannable ( k — 1 )-cycle associated with g is or is not bounding below c.
In verifying the corollary in case k > 0 one notes that a linking fc-cycle is
associated with g according as 7*_i is or is not bounding below c. In case 7*_i
is not bounding below c, 7^-1 is what we have called a newly-bounding ( k — 1)-
cycle associated with g , and in this case ARk-i = — 1. If fc = 0, there are no
linking or newly-bounding cycles, and just on^ critical fc-cycle in maximal sets
of such cycles. In this case A Rq = 1.
We term the extremal g of increasing or decreasing type according as A 2?* = 1 or
A/2*-i = — 1.
From Theorem 10.2 we now obtain the following important consequence.
230
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
Theorem 11.4. If all the critical extremals are non-degenerate , the number Nk of
distinct extremals of index k is such that
Nk ^ Pk ,
where Pk is the kth connectivity of the unrestricted functional domain il.
A problem in which all the critical extremals are non-degenerate will be called
non-degenerate . In the next section we shall prove that the non-degenerate
problem is the general problem, at least in two important classes of problems.
The connectivities Pk are topological invariants of the Riemannian manifold R
and the manifold of end points Z. One can frequently determine these con¬
nectivities by a study of a particular extremal problem defined on R . We
formulate this result in the following corollary.
Corollary. If corresponding to a given Riemannian spare R and terminal
manifold Z, there exists an integral J defined on R such that all the critical extremals
are non-degenerate and of increasing type , then the connectivity I\ of the functional
domain Q equals the number of distinct extremals of index k.
The non-degenerate problem
12. It follows from Theorem 11.1 that a non-degenerate critical extremal g is
isolated from other critical extremals in the sense that there exists no other
critical extremal with an arbitrarily small /-distance from g. It then follows
that the number of critical extremals with /-lengths less than a constant b is
finite in a non-degenerate problem. For otherwise the initial points and direc¬
tions of such critical extremals would have a cluster point and direction (P, X) on
R, and the extremal through P with direction X, taken with a suitable limiting
length, would be a critical extremal which was not isolated, and hence would be
degenerate.
Accordingly in a non-degenerate problem the critical extremals are either finite in
number, or else form a countably infinite sequence of extremals
Oh Oh
whose J-lengths become infinite with their subscripts.
We have seen in Ch. VI that in the case of an analytic function f{xh • • • , xm)f
defined over a region in euclidean space, the non-degenerate case may be regarded
as the general case, in that / can always be approximated arbitrarily closely by an
analytic function whose critical points are non-degenerate. We shall establish
the corresponding fact for functional problems, at least in case the problem has at
most one variable end point. The proofs are necessarily more difficult than in.
the case of a function/, because in the case of a functional problem there are in
general infinitely many critical extremals, and in approximating such a problem
these critical extremals must all be replaced by non-degenerate critical extremals.
We begin with the fixed end point problem. Let P be a point on R. Let g
be an extremal issuing from P. Let the extremals through P with initial direc-
112]
THE NON-DEGENERATE PROBLEM
231
tions neighboring that of g be represented neighboring P as in Ch. V, §5, in the
form
(12.1) x ' — x'(t, u ),
where t is the arc length along the extremals measured from P, and ( u ) is a set
of n — m — 1 parameters chosen as in Ch. V, with ( u ) — ( u0 ) defining g. Let the
extremal determined by ( u ) be denoted by g(u).
Let t = to represent a conjugate point Q of P on g(u0) and let ( z ) be a set of
local coordinates on R neighboring Q. Let the continuation of the extremal g(u)
be represented, neighboring Q, in the form
(12.2) - h'(t,u\
where t is the arc length along g(u) measured from P. In (12.2) t is confined
to values near to. As in (12.1) the parameters (u) neighbor (u0). We shall
confine the sets (/, u) to a closed convex neighborhood N of the sets (to, w0) on
which the functions (12.2) are analytic. The conjugate points of P on the
ext remals g(u) for (/, v) on Ar are given by the zeros of the jacobian
(12.3)
D(t, u)
D(ty U\ * *
zm)
Tun
(n = m — 1)
provided A' is sufficiently small.
We shall prove the following lemma.
Lemma 12.1. In the space ( z ) the conjugate points of P on the extremals g(u), at
points (t, u) 07i Ar, form a set whose measure is null .
In proving this lemma it will be convenient to set
(t, ul, • * • , Un) = (vl, • • • , vm) = (v)
and (t0f u0) = (r>0). We also write (12.2) in the form
2' = h'(t, u) = z'( v)
and set
D(t , u) = A(v).
Let (a) be any set (v) such that A(a) = 0. The rank r of the determinant
A(u) will lie between 0 and m exclusive. In the space (z) let the r-plane
(12.4) = z'(a) + (vh - a*) (i, h = 1, • ■ • , m)
l x1 denoted by a>a. The distance d(v, a) of the point zl(v ) from the /-plane o>a
will be at most the square root of the quantity
(12.5) ^ £r'(i>) - ■?<(«) - —f- (.v* -a*)J = 2) [bL(v, a) (vh - «*) (vk - a*)]*.
232
THE BOUNDARY PROBLEM IN THE LARGE
[VII ]
The bracket on the right is obtained by applying Taylor’s formula with the re¬
mainder as a term of the second order to each of the m differences appearing in
the bracket on the left. The coefficients blh(v, a) are accordingly less in absolute
value than some constant independent of the choice of (a) and ( v ) on N. If
we set
p2 = — a*) ( — ah) (h = 1, • • • , m),
taking p itself as positive, we see from (12.5) that
d(v, a) ^ kp 2
where k is a positive constant independent of the choice of (v) and (a) on N.
Let D(v , a) be the distance in the space ( z ) from the point z'(v) to the point
z *(a) . One sees that
D(v , a) ^ Xp
where X is a constant independent of ( v ) and (a) on N.
Let si be a region in the space (t>) consisting of the points (t;) interior to and on
an ( m — l)-sphere of radius a with center at (a). Suppose A(a) = 0. Restrict¬
ing points ( v ) to N, the points z'(v) corresponding to points (t>) on saa will be
included in a region v ea in the space ( z ) consisting of the points (z) on wa at most a
distance \a from the point [z(a)], together with the points on perpendiculars to
03a at most a distance k<r 2 from the points (z) already chosen on o>a. This follows
from the choice of X and k.
To determine a useful upper bound for the volume v*a of v°a let the space ( z )
neighboring [2(0)] be referred to rectangular coordinates (z), with the origin at
(2) = (a), and with the coordinate axes of x1, • • • , xr in the r-plane o)a. It
appears then that v°a will lie in the rectangular region
I xh I g \a (h = ],•••, r),
(12.6)
| xk | 5* kc* {k — r + 1, • • • , m).
The volume v*a will be less than the volume of the region (12.6), that is,
(12.7) vl < \rarkm~rcr2(rn~r) = \rkm-r<rmam~r.
Let 6 be an arbitrarily small positive constant. Let si represent the volume of
the region si in the space ( v ). From (12.7) we infer that there exists a positive
constant p« independent of (a) on N, and so small that, when a g pe,
(12.8) vl < esl
for all points ( v ) = (a), for which A (a) = 0.
Let us now cover N in the space ( v ) by a set of non-overlapping congruent
*n^-cubes K with diameters at most p,. We prefer those ra-cubes which contain
points ( v ) = (a) of N at which A(a) = 0. Let each preferred cube K be included
in a spherical region si such that (a) lies on K , A(a) = 0, and <r is as small as is
112]
THE NON-DEGENERATE PROBLEM
233
consistent with si containing K . It appears that a will then be at most the
diameter of K , so that if K denotes the volume of the m-cube K we have
(12.9) si < ixK
where p, is a numerical constant depending only on the dimension m. Combin¬
ing (12.9) and (12.8) we have the result
(12.10) vl < epK .
With each preferred ra-cube K in the space ( v ) we have then associated a
region vl in the space ( z ) containing all of the points [z(v)] for which ( v ) is on the
intersection of K and N. A set of regions v°a which includes a region v°a for
each preferred m-cube K will contain all conjugate points \z(v)] of P for which
(i>) lies on A7. The total volume V of this set of regions vaa will be such that
V < ejuZK,
where the sum 2 extends over the preferred m-cubes. Now 2K is bounded re¬
gardless of the diameters of the cubes K, p is fixed, and e is arbitrarily small.
Hence V is arbitrarily small.
The lemma follows directly.
We shall now prove the following theorem.
Theorem 12.1. The set of 'points on R which are the conjugate points of a fixed
point P has the measure sero on R.
Recall that the volume of an elementary region on R is given by the invariant
integral
/ • • ' / I 9ii(x) I ■■■ dxm.
From the preceding lemma it then appears that the set of conjugate points
[z(t>)] defined by the vanishing of A(v) in the neighborhood N in the space ( v )
has a measure zero on R . This is a result in the small. To proceed we need a
representation of the extremals through P as a whole.
Let ( x ) be ft coordinate system containing P. Let (a) be the set of direction
cosines in the space (x) of a ray issuing from P. Let the extremal issuing from
P with the direction (a) be denoted by 7(a). Let the point on y(a ) at a distance
t > 0 from P along g on R be represented by ( t , a). The sets (t, a) can be repre¬
sented as points on a domain 2 which is the product of the interval 0 < t < <*>
and the unit (m — l)-sphere
+ ‘ ‘ * + «m = L
If the extremal 7 (a) is identical with the extremal determined by the param¬
eters (u) in (12.3), we say that the set ( t, u) in (12.3) represents the point
(t, a) on 2. The sets (U v) = ( v ) which lie on the neighborhood N in the space
(u) thus represent the neighborhood of a point on 2. Moreover the neighbor¬
hood of each point on 2 can be similarly represented in such a fashion that for
234
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
each neighborhood the points [2(*>)1 which are conjugate to P have a measure
zero on R. The points (tf a) on 2 for which t lies between two finite positive
constants can be included on a finite set of such neighborhoods. All the points
on 2 can accordingly be included on a countably infinite set of such neighbor¬
hoods. The set of conjugate points of P on R can thus be included in a count¬
ably infinite set of regions of the type of vaa in such a fashion that the sum of the
volumes on R of the regions vaa is arbitrarily small.
The proof of the theorem is now complete.
A study of the geodesics on a torus would disclose the fact that there are
points P on the torus, the set of whose conjugate points on the torus is every¬
where dense. In spite of this fact the preceding theorem gives us the following
corollary.
Corollary. The set of points Q on R which are conjugate to a fixed point P on
no extremal through P is everywhere dense on R.
A pair of points P and Q of which P and Q are mutually conjugate on no
extremal joining P to Q will be termed non-degenerate. All other pairs will be
termed degenerate. It follows from the preceding corollary that if P and Q are
a degenerate pair, an arbitrarily small and suitably chosen displacement of Q or
P will replace P and Q by a non-degenerate pair. We see that the non-degen¬
erate pair and corresponding non-degenerate problem may properly be considered
as representing the general case.
In §14 we shall prove similar theorems concerning focal points of a manifold.
The fixed end point problem
13. This section presents a study of the most important special problem in the
large. In it we not only obtain precise results in the non-degenerate case,
but also show how the degenerate case may be treated as a limiting case of the
non-degenerate case.
As a matter of notation it will be convenient to denote the functional domain Q
corresponding to two fixed points A \ A2 by
(13.1) 12 (A\A2).
We have seen in Theorem 11.3 that the type numbers of a non-degenerate
critical extremal g of index k are all null except that Mk = 1. In the case of the
fixed end point problem we have also seen in Ch. Ill that the index of g is the
number of conjugate points of A1 on g between A1 and A 2. The theorems of
§10 can now be re-interpreted for a non-degenerate fixed end point problem. In
particular Theorem 10.2 gives us the following.
Theorem 13.1 . If A\ A2 is a non-degenerate pair of points on R> the number of
extremals g joining A1 to A2 with k conjugate points of A1 on g between A1 and A 2
must be at least as great as the kth connectivity Pk of the functional domain Q(Al, A2).
THE FIXED END POINT PROBLEM
235
[ 13]
We shall show how this theorem leads by a limiting process to a theorem appli¬
cable both to the degenerate and to the non-degenerate case. To that end we
turn to the extremals represented in (12.1). We denote the extremal in (12.1)
determined by (?^) by g(u). We shall use the following lemma.
Lemma 13.1. If on the extremals (12.1) through the point P, the value t = to
determines the kth conjugate point of P on g(uo), the kth conjugate point of P on the
extremal g(u) exists for (u) sufficiently near ( u0 ) and lies at a distance , t(u),fromP
along g(u, ), which varies continuously with (u). If on the other hand t = to is not
the kth conjugate point of P on g(uj)y then the kth conjugate point of P on g(u ), if it
exists, will not he determined by values of t near to, provided (u) is sufficiently near
0/i>).
It is understood that conjugate points are counted with their indices.
To prove the first statement of the lemma, suppose t = to is the A*th conjugate
point of P on g(u 0). Let t ' and t" be t wo values of t such that
t" < to < t\
and such that V and t" separate /(, from the other values of t which define con¬
jugate points on g(uo). On g(u) we take A1 and A2 respectively as the point P
and the point Q(u) on g(u) at which t ■= //. For these end points we then set up
the index form corresponding to g(u) essentially as in §14, Oh. V.
More explicitly we cut across the extremal g(uo) with the intermediate mani¬
folds Mq of ( "h. V, and choose the parameters (v) of Ch. V, as the ensemble of the
successive sets of parameters (/ 3) determining points on the intermediate mani¬
folds Mq. We evaluate J along the broken extremal whose end points are P
and Q(u) and whose intermediate vertices are on the respective manifolds Mq
at the points determined by ( v ), restricting (v) to sets near (0). We denote the
resulting function by J(v, u). We see that g(u) will meet the manifold Mq
in a point (j8) whose parameters l h , will be functions fih(u) of class C2 in the
variables (u) for (u) near (w0). We denote the ensemble of these successive sets
\&(u)\ by [c(?/)J, and define the index form corresponding to g(u) as the form
d2 J
P(z, M) = C’(m), u)z'z> (i, j = 1, , np).
dv'dvJ
The form P(z, u) is non-singular for (u) = (w0), since Q(u0) is not a conjugate
point of P on g(u{)) for t = For a sufficiently small variation of ( u ) from (w0)
the form P(z, u) will remain non-singular and unchanged in index. But its
index is the number of conjugate points of P on g(u) for which 0 < t < tf.
Thus the kih conjugate point of P on g{u) must exist and lie at a distance t(u)
on g(u) from P such that t(u) < t'.
We now prove in a similar fashion that t(u) > l” . Since V and t” can be taken
arbitrarily close to t = /0 it follows that t{u) must be continuous at (w0)- But
t(u) is similarly shown to be continuous at other nearby values of (u), and the
proof of the first statement of the lemma is complete.
236 THE BOUNDARY PROBLEM IN THE LARGE [ VII ]
The second statement of the lemma follows from a similar use of the index
form.
We shall show that the connectivities of Q(P, Q ) are independent of the choice
of P, Q on R . We begin with a definition.
The extension of a curve g by curves p and q. Let p, g , and q be continuous
images on R of the line segment 0 S t g 1. Suppose p, g> and q can be joined
in the order written to form a single continuous curve g*. We assign a param¬
eter r to the points of g* such that 0 ^ r ^ 1, and such that the variation of r
corresponding to any segment 7 of g* is proportional to the sum of the variations
of the parameters t on the segments of 7 on p, gf and q. We term g* so parameter¬
ized the extension of g by p and q.
The extension of aj-cycle on 12 by curves p and q. Let a» be an i-cell on 12(P, Q)t
the image on R of a functional simplex c* X h. Let p and q respectively be sensed
curves on R which join a point P' to P and Q to a point Qr. Let g be any curve
of ai represented by a product o> X ti of a point w on c* and the line segment
0 ^ t S 1. We “extend” g by p and q as in the preceding paragraph, forming
thereby a curve g* with parameter r. We represent g* by the product 0) X r 1.
We thereby obtain a new z-cell a* on 12(P', Q'), an image on R of the functional
simplex c< X t\. We term a* an extension of a, by p and q .
The extension of an z-chain Wi on 12 by p and q will now be defined as the chain
obtained by extending each of the cells of w , by p and q. It will be observed
that the extensions of cells on 12 which are conventionally identical will again be
conventionally identical.
Let g be the continuous image on R of the line segment 0 g t ^ 1. Let U
represent any point on R. We suppose that we have given a deformation T of q
of the form
(13.2) U = U(t, M)
where U is a continuous point function of the curve parameter t and the time
g for
0 g t S 1,
(13.3)
0 5a ft ^ 1,
where (13.2) defines g when p = 0. Under T we understand that the point
V ( t , 0) on g is replaced at the time ft by the point U (t, p). Under T the curve g
is deformed into a curve g' on which
U « U(t, 1) (0 g t g 1).
Let p be the curve traced by the initial end point of g under T, taking the time
ft as the parameter. Let q be the curve traced by the final end point of gf under
the inverse of T} taking ft' = 1 — ft as the parameter. The curve g ' “extended”
by p and q affords a sensed curve g* joining the end points of g on R. On g*
the parameter is r, with 0 ^ t g 1.
[13]
THE FIXED END POINT PROBLEM
237
It is clear that g can be deformed into g*, holding its end points fast. We
need however to make such a deformation more explicit, and in particular to
show that it can be so defined as to be determined completely by the preceding
deformation T.
We accordingly introduce a deformation Tf of g into g*, deriving Tf from T as
follows.
In the {t, ju)-plane consider the square (13.3). In this square the side
(13.4) 0 g t ^ 1, M = 0,
represents g , while the three remaining sides may be regarded as a curve y
representing g*. To define T' we join each point t of the segment 0 ^ t g 1 of
the t axis by a straight line to that point of y which represents the point on g *
at which r = t. We let the points ( t , m) on (13.4) move along the resulting
straight line segments at rates which equal the lengths of these segments. The
corresponding points U ( t , n) on R will move on R so as to define a deformation
T ' of g into g*. Under Tr the end points of g remain fixed. During Tr we assign
the same parameter t to the moving point as it initially possessed on g.
We continue with the following lemma.
Lemma 13.2. Let Wj be a j-cycle on 12(P, Q ) which is non-bounding on 12(P, Q).
Suppose that there exists a deformation T which deforms P and Q in a unique manner
into points P' and Q' on Ry and deforms w} into a j-cycle Zj on 12(P' , Q'). The
cycle Zj will be non-bounding on 12(P', Q').
Let p be the curve traced by P under T and q the curve traced by Q' under the
inverse of T . Let E{zj) represent the extension of Zj by p and q. The cycle
E{Zj) will lie on 12(P, Q ). We now apply the deformation T' “derived” from T
to each curve of Wj. We thereby obtain a deformation of Wj into E(zj)} holding
P and Q fast. We thus find that
(13.5) Wj ~ E(zj) [on 0(P, Q)}.
If the lemma were false, Zj would bound a chain Zj+ \ on £2(P', Q')* Let
E(zj+i) be the extension of z,+i by p and q . We have
(13.6) E(zj+ 1) -> E(zj) [on fl(P, Q)],
so that E(zj) ^ 0 on Q (P, Q) . Hence
Wj ~ 0 [on 12 (P, Q)],
contrary to hypothesis.
We infer the truth of the lemma.
We shall now prove the following lemma.
Lemma 13.3. Suppose a j-cycle Wj on Q (P, Q) has been extended by curves p and q
to form a j-cycle Zj on Q (Pf, Qf ). If Wj is non-bounding on Q (P, Q)y z, will be non¬
bounding on 12(P', Q').
238
THE BOUNDARY PROBLEM IN THE LARGE
l VII J
Let g be any curve of Let g* denote the curve obtained by extending g by
p and q. We shall now define a deformation of ivj into z}. In it we let each
point r, initially on g} move along g* to the point on g* which possesses a param¬
eter r equal to the parameter t initially possessed by V on g. We let V thereby
move so that its image on the r axis representing g* proceeds at a rate equal to
the length of the segment of the r axis to be traversed. One thereby deforms
into Zj. By virtue of the preceding lemma z, is non-bounding on tt(P\ Qf).
We now come to a theorem of purely topological content.
Theorem 13.2. The connectivities of the functional domain i l(Pf Q) are inde¬
pendent of the choice of the points P} Q.
Let P, Q and P', Qf be two pairs of points on R . Let p and q be respectively
two sensed curves which joint P' to P and Q to Q'. Let w7 be any j-cycle on
il(P, Q). Let tOj be “extended” by p and q to form a cycle Zj on il(P\ Q'). By
virtue of the preceding lemma z} will be non-bounding on tt(P', Qr) if w} is non¬
bounding on Q(P, Q). It follows that the jth connectivity of i l(P'f Qr) must be
at least that of i~l(P> Q ). Upon interchanging the roles of P, Q and Pf Q' one
sees that the connectivities of 1I(7>, Q) and ll(P’, Q') must be equal.
The tivo point connectivities of R. Since the connectivities Pk of il(P, Q) are
independent of the choice of points P , Q on /f, we can properly omit reference to
P and Q and term Pk the Alh two point connectivity of R.
The function J k(P , Q). Suppose the At h two point connectivity of R is not
zero. Corresponding to any two points P and Q on R we let
JdP, Q) (A = o, l, •■ •)
be the inferior limit of constants c such t hat there is at least one restricted A-cycle
which is non-bounding on il(P, Q ) below c. We continue with th(‘ following
lemma.
Lemma 13.4. The function J k(P, Q) is continuous in P and Q for arbitrary
choices of P and Q on R.
Let P, Q and P'y Q' be two pairs of points on R. Let | XV | denote the
./-distance between points X and Y on R. Suppose that
(13-7) 1 PP’ | < e, \QQ'\<e,
where e is an arbitrarily small positive constant. By virtue of the definition of
Jk(Pj Q) there exists a restricted A-cycle wk on tt(P, Q) which is non-bounding on
S 2(P, Q) and on w hich
J < J k(P , Q) + e.
Let p and q be two sensed elementary extremals joining Pf to P and Q to Qr
respectively. Let u/k denote the A-cycle on il(Pr, Qf) obtained by extending wk
H3]
THE FIXED END POINT PROBLEM
239
by p and q . On wk we see that the maximum of J exceeds the maximum of J
on wk by at most 2e. Hence on w'k,
J < Jt(P, Q) + 3e,
so that
Q') < MP, Q) + 36.
But wc can reverse the roles of P, Q and P'y Qf and hence conclude that
I Jt(P', Q’) - JUP, Q) | < 3c,
subject to (13.7)
The lemma is thereby proved.
We shall now prove a general theorem.
Theorem 13.3. If the kth two-pond connectivity of R is not zero, there exists an
extremal gk joining an arbitrary pair of points J\ Q on R , with the following proper¬
ties. The J -length of gk is Jk(P, Q) and vanes < ontinuously with P and Q on R.
If P, Q is a non-degenerate pair of points , there are exactly k conjugate points of
P on gk . If P} Q is a degenerate pair of points , there are at least k and at most
k + w - 1 conjugate points of P on gL including Q.
Suppose first that the pair P, Q is non-degenerate. Set J/fP, Q) = c. The
number c must then be a critical value of ./, and among the extremals of /-length
c there must be at least one extremal g for which the number Mk — 1. Otherwise
if cl and c" were two constants not critical values of J separating c from other
critical values of «/, with cf < c < c"} every restricted ^-cycle of tt(P, Q ) for
which J < c" would be homologous on U(P, Q) to a A-cycle on J < c'f so that c
could not equal JifP, Q).
In case Py Q is a degenerate pair, let
a - i, 2, •..)
be a seqiR'nce of points tending towards Q as i becomes infinite, and such that
the pairs P, Qt are non-degenerate and distinct. Let y ,• be an extremal satisfy¬
ing the theorem for the pair Pf Q> and for the given k. Let \i represent the initial
direction of yt at P. The directions Xt will have a cluster direction X at P and
the extremal g issuing from P with the direction X will reach Q after traversing a
./-length J),(Py Q).
Suppose there are exactly h conjugate points of P on g including Q. It follows
from Lemma 13.1 that h k. But there will be at least h — (m — 1) conju¬
gate points of P on g excluding Q. By virtue of Lemma 13.1 there will t hen be at
least h — (m — 1) conjugate points of P on each extremal y i with initial direc¬
tions sufficiently near X. But the number of conjugate points of P on Yt is
exactly k so that
k ^ h — (m — 1).
240
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
It follows that
h ^ k + m — lr
and the proof of the theorem is complete.
It would be a mistake to believe that the extremals 0* affirmed to exist in the
preceding theorem could always be chosen so as to vary continuously with their
end points P, Q. Simple examples show that the contrary is true.
The case P — Q presents no special difficulty. If however P = Q and k = 0,
the extremal 0* of the theorem reduces to a point. For k > 0 the extremals 0*
exist and possess the same properties as when P ^ Q.
The one variable end point problem
14. We here suppose that the second end point A 2 is fixed at a point Q and
that the first end point rests on a non-singular analytic (m — l)-manifold Af,
the image on R of an auxiliary simplicial circuit B.
We suppose that M is orientable . To define this term let us understand that a
covariant (or contravariant) vector at a point P on R varies continuously with P
if its components in each local coordinate system vary continuously with P.
Starting with a point P0 on M and a unit covariant vector X° normal to M at P0,
let P vary continuously along a path on M which starts and ends at P0, and let a
unit covariant vector X normal to M at P vary continuously with P starting from
the vector X°. If X returns to X° no matter what the path, M is termed orientable.
If M were non-orientable, we could replace M by an orientable covering manifold
and proceed in essentially the same way.
We begin with a study of the transversality conditions in the large. Let (x)
be a local coordinate system and (x) = (a) a particular point (x). Let r* be the
components of a unit contravariant vector defining a direction at (x) = (a).
Let Xi be the components of a unit covariant vector orthogonal to M at the
point (x) = (a). A necessary and sufficient condition that the extremal tangent
to { r ) at (x) = (a) cut M transversally at (x) = (a) is that there exist a constant
fi such that
(14-1)' Fri(a, r ) - /xX, =0 (i = 1, • • • , m),
(14.1) * 0i,*(a)rlry = 1.
In the search for all directions (r) which cut M transversally at (x) ~ (a) we
lose no generality if we impose the condition,
(14.1) "' M > 0.
For if (14.1)' and (14.1)" are satisfied we have
(14.2) /xXir* = rlPr»(a, r) = F(a, r) > 0,
241
[ 14 ] THE ONE VARIABLE END POINT PROBLEM
We will then either have
(14.3) X»r' > 0
and hence m > 0, or else upon replacing A» by X' = —A*, and /i by /i' = we
will have
Fri(a, r) - m'X • = 0
with (14.1)^ satisfied, and u > 0 as before.
We shall now prove the following lemma.
Lemma 14.1. The conditions (14.1) define a one-to-one analytic correspondence
between the totality of unit contravariant and covariant vectors (r) and (X) at the
point (t) = (a).
If a unit contravariant vector (r) is given, conditions (14.1) uniquely determine
the components of a unit covariant vector (X) as analytic functions of the
components of (r).
On the other hand suppose sets (r), ju, (X) are initially given satisfying (14.1),
with (X) a unit covariant vector. One can then vary the vector (X) independ¬
ently among unit covariant vectors, and solve the system (14.1) for the varia¬
bles (r) and n as analytic functions of the components of (X). For the jacobian
of the system (14.1) with respect to the variables (r) and m is seen to be
F riTj A,
(14.4) = 2F1ri\i,
2gijri 0
and this is not zero by virtue of (14.2), and the hypothesis that F i ^ 0. Thus
the relation between the vectors (r) and (X) defined by (14.1) is locally analytic
and one-to-one. But one can continue this correspondence by varying (X)
subject to the condition
(14.5) 0«X<A/ = 1.
Regarding (X) as a point, condition (14.5) requires (X) to rest on an (m — 1)-
ellipsoid. It follows from the topological properties of an ellipsoid (or ( m — 1)-
sphere) that the above correspondence is one-to-one in the large.
Let (X) and ( — X) be the two unit co variant vectors normal to M at ( x ) = (a).
According to Lemma 14.1 there are two unique contravariant vectors (r) and (f)
which satisfy (14.1) with (X) and ( — X) respectively. Upon using (14.2) we see
that the sums
A,rV X if1
have opposite signs, a property which makes it possible to distinguish between
r* and f\
The unit contravariant vector r(P). Corresponding to each point P on M let
A(P) be a unit covariant vector normal to M a.t P and so chosen as to vary con-
242
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
tinuously with P on M. Such a choice of X(P) is possible since M is orientable.
Corresponding to X(P) let r(P) be the unique unit contravariant vector r(P)
which cuts M transversally at P and which is chosen from the two vectors which
cut M transversally at P by requiring that
(14.6) X;r‘ > 0,
where Xt and rl are the local components of X(P) and r(P) respectively.
Let M be regularly and analytically represented in the coordinate system
(a;), neighboring a point (x) = (a) on M , by functions
(14.7) = xi{u11 • * • , un) (n = m — 1),
where x*(w0) = a*, and the parameters (u) neighbor ( u0 ). We shall now prove
the following lemma.
Lemma 14.2. In the system ( x ) the components rl of the contravariant vector r(P )
are analytic functions r'(u) of the parameters (u) locally representing M.
First recall that the components Xt of the preceding covariant vector X(P)
normal to M at P are analytic functions X,(?z) of the parameters (u) of P, for (u)
neighboring (u0). We now write (14.1) in the form
(14.8)
Frt(x{u), r) - ai\,(u) =0 (ju > 0),
= 1.
As previously we see that we can solve for the variables r* as analytic functions
r* = R'(u)
of the variables (u) for ( u ) near (w0). The solution Rl(u) so obtained satisfies
the condition
R\u)\i(u) > 0
as follows from (14.8). But the vector r(P) satisfies the same conditions at the
point (u) as the vector R\u)} and is thereby uniquely determined. Thus
r'(u ) = R'(u)
neighboring ( u0 ).
The proof of the lemma is now complete.
The unit contravariant vector f(P), Corresponding to each point P on M let
f(P) denote the unit contravariant vector which cuts M transversally at P and
in terms of the components X, of the preceding vector X(P) satisfies the condition
\iP < 0.
One readily proves that the contravariant components of f(P) are again analytic
functions f ((u) of the local parameters (u) of M.
We now parallel the results in §12 on the measure of the conjugate points of a
fixed point.
[14]
THE ONE V ARIABLE END POINT PROBLEM
243
We represent M locally as in (14.7). Let g(u) be the extremal issuing from the
point x!’(w) on M with the direction r^u) of Lemma 14.2. Let t be the distance
along g(u) from M. Let Q be a focal point of M on g(u0) with t — t0. Let (z)
be a set of local coordinates of M neighboring Q. Neighboring Q the extremal
g(u) can be represented in the form
?/).
We can prove as under Lemma 12.1 that the focal points of M on the extremals
g(u) for t near to have a measure in the space (z) which is null. We are then led
to a similar result concerning the extremals g(u ) issuing from the point (u) on
M with the directions fl(u).
Theorem 12.1 is here replaced by the following theorem.
Theorem ILL The set of focal points of M on It has a measure on It which is
null.
The proof of this theorem can be given essentially as was the proof of Theorem
12.1. The representation in the large of the extremals cutting M transversally
is necessarily somewhat different.
To obtain such a representation let g(fy) and g(P) be respectively the ex¬
tremals issuing from the point P on M with directions r(P) and f(P). The point
on g(P) at a distance t from M on g(P) will be represented by the pair (Pf t).
The sets (P, /) form a domain Z representable as the product, M X t*, of M and
the interval
l*: - oc < t < oo.
With the aid of Z one can prove as in §12 that the focal points of M belonging
to the extremals g(P) have a measure zero on R. One can then establish a
similar result for the extremals g{P), and thereby complete the proof of the
theorem.
We state the following corollary.
Corollary. The set of points on R which are not focal points of M is everywhere
dense on R.
Focal points of M belonging to the extremals g(P) or g{P) at points at which
t > 0 will be called positive focal points of M} white focal points at which t < 0
will be called negative focal points of M. Positive focal points are relevant in the
problem in which the first end point rests on M and the second is fixed, while
negative focal points are relevant when the first end point is fixed and the second
end point rests on M .
Let 12(Af, Q) denote the functional domain Q in the problem in which the first
end point rests on M and the second end point is fixed at Q. Theorem 13.1 is
then replaced by the following.
Theorem 14.2. If Q is not a positive focal poiiit of M nor on M, the number of
extremals g which join M to Q, which cut M transversally at their initial points and
THE BOUNDARY PROBLEM IN THE LARGE
244
[VII]
possess k positive focal points of M thereon between M and Q , is at least as great as
the kth connectivity of the functional domain Q(M, Q).
We can prove that the connectivities of Q(M , Q) are independent of the choice
of Q among points Q on Ry following the analogous proof in §13. One uses the
extension of a curve g by curves p and q as defined in §13. In the present case
we need q only, and take p as null. We record the theorem as follows.
Theorem 14.3. The connectivities of the functional domain Q(M, Q ) are inde¬
pendent of the choice of Q on R.
The function Jk(My Q). Suppose the fcth connectivity of Q(M, Q ) is not
zero. Let
MM, Q)
be the inferior limit of constants c such that there is at least one restricted fc-cycle
which is non-bounding on Q(M, Q) below c. We can prove, essentially as in
§13, that the function Jk(My Q) is continuous with respect to a variation of
Q on R.
We come finally to the analogue of Theorem 13.3.
Theorem 14.4. If the kth connectivity of tt(M, Q ) is not zero and Q is not on M,
there exists an extremal gk which joins M to Q with the follovnng properties.
The extremal gk cuts M transversally at its initial point on M. The J -length of
gk is Jk{My Q ) and varies continuously with Q. If Q is not a positive focal point
of My there are exactly k positive focal points of M and gk on gk between M and Q
exclusive. If Q is a positive focal point of M and gkf there are at least k and at most
k + m — 1 positive focal points of M and gk on gk including Q.
The proof of Theorem 14.4 is practically identical with that of Theorem 13.3
and can be omitted.
One can admit that Q lies on M if one understands that go reduces to a point
in that case.
The two point functional connectivities of an m-sphere
15. We shall now suppose that the Riemannian space R is the topological
image of a unit ra-sphere Sm in a euclidean ( m + l)-space (w) and that the
terminal manifold is represented by two distinct points A1 and A2 on Sm. We
shall determine the connectivities Pk of the corresponding functional domain
A2).
We have seen in §13 that the connectivities Pk of tt(Aly A2) are independent
of the choice of the points A1, A2 on R. Without loss of generality we can take
A1 and A2 at the interesections of Sm with the positive w\ and w 2 axes. More¬
over the connectivities Pk are independent of the particular image R of Sm
which is chosen. We can accordingly take R as Sm itself. We shall suppose
moreover that J is the integral of arc length on Sm and shall make use of the
Corollary of Theorem 11.4 to determine the connectivities P*.
[ 15 ] TWO POINT FUNCTIONAL CONNECTIVITIES OF m-SPHERE
245
Let A be the point on Sm diametrically opposite to A1. The extremals
joining A1 to A2 on Sm are arcs of great circles simply or multiply covered. We
naturally regard two extremals as distinct if they overlap each other, but are
not identical in length. Taken in the order of their lengths, our critical ex¬
tremals are then an infinite sequence of geodesics
0o, g h • • * >
of which g0 is the arc of the great circle of length less than t joining A1 to A 2,
and is the residue of the same great circle. The arcs go and gx have opposite
senses relative to their common great circle 7. The geodesic g2r joins A1 to A 2
on 7, agreeing in sense with g0 and passing A r times. The geodesic 02r+i joins
A1 to A2 on 7 , agreeing in sense with gh passing A r + 1 times.
To apply the theory of the non-degenerate extremal we need to know the
index of gp. Observe first that gp is non-degenerate since its end points are not
conjugate on gp. Its index equals the number of conjugate points of A1 on gp at
distances from A1 on ^ less than the length of gp. Each conjugate point P of
A1 is thereby counted a number of times equal to its index. The index of a
conjugate point P on gp is the number of linearly independent solutions of the
Jacobi equations corresponding to g which vanish at the points tl and t 2 on the t
axis corresponding to A1 and P. Now the conjugate points on *Sm are diametri¬
cally opposite points so that every extremal through A 1 passes through P. Con¬
sequently every secondary extremal vanishing at t1 must also vanish at t 2. The
index of each conjugate point of A1 is accordingly m — 1, and the index k of gp
is thus seen to be
(15.1) k = p(m — 1).
For, starting from A1, gp passes A and A1 in turn, until p passages have been
made all told.
We state the following lemma.
Lemma 15.1. The geodesics g0y g 1, • • • are of increasing type in the sense of §11.
This is true of go by virtue of the Corollary of Theorem 11.3, since go is non-
degenerate and has the index zero.
To prove the lemma for g 1, g2y • •• we shall associate a restricted cycle X* with
gp, and show that X* is a linking cycle corresponding to gp. Here k is the index
of gp and equals p(m — 1).
Corresponding to gP} p > 0, we introduce a set of p constants
e, (? = !,•••, v).
such that
(15.2) 0 < ei < e2 < • • • < ep < 1.
In the space ( w ) let
M“ (q = 1, • ■ • , v)
246
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
be an (to — l)-sphere formed by the intersection with Sm of an m-sphere S£
with radius eq) and center at A if q is odd, and at A1 if q is even. Let Pq be an
arbitrary point on Mq. No point on any of the spheres Mq is diametrically
opposite to a point on any other such sphere, or to A1 or A2, as follows from the
choice of the radii eq. Hence the points
(15.3) A\ P\ • • * , Ppy A2
can be successively joined by unique minimizing arcs of great circles. The
points (15.3) will be regarded as the vertices of a point (x). If the constant p of
§2 limiting the lengths of elementary extremals is taken near enough to the
number x, these points (7r) as well as the broken geodesics g(v) will be admissible.
We suppose g{r) represented in terms of a parameter t proportional to the arc
length measured from the initial point of g(ir) and running from 0 to I. The
totality of the above points (x) defines a cycle ck representable by the product
Tk = A3 X Ml X M‘L X • • • X Mp X A2.
We note that the dimension k of ck is given by the equation
k = p(m — 1),
and equals the index of gp.
The totality of the curves g( x) determined by points (x) on l\ forms a re¬
stricted fc-cycle X/c on U as we shall now see.
Suppose Tk has been subdivided into cells so that it may be regarded as the
sum of a set of closed /c-cells ak which are the images of closed siinplices a*. Let
(x) be a point on ak and P the corresponding point on ak. We represent the point
t on g(ir) by the pair (P, /), thus representing g(w) by the product P X t\ of P
and the line segment L:0 g / g 1. The ensemble of curves g(w) determined
by points (x) on ak will thus be represented by the closed functional simplex
ak X t\. In this way we see that the totality of curves ^(x) determined by points
(x) on T* can be represented as the curves of a restricted fc-cycle \k on to.
We continue with a proof of the following statement.
(A). The integral J assumes an absolute , proper maximum over on the
curve gp.
Set k — p(rn — 1). The vertices of gp as a curve of X* are respectively A1,
the p successive first intersections of gp with the m-spheres Mqf and finally the
point A2. Let hh • • • , hp+i be the corresponding set of elementary extremals
making up gp. Denote the length of hq by hq. We see that
h\ = x — eif
ht * x — {ex — et),
(15.4) .
hp = x — ( ep — ep-i).
K+i * */2 + «p.
[ 15 ] TWO POINT FUNCTIONAL CONNECTIVITIES OF 772-SPHERE
247
That gp has the maximum length among curves of Xfc follows from the fact that
on any broken extremal of \k the respective elementary extremals have at most
the lengths of the corresponding elementary extremals in (15.4).
Let 7 be any broken extremal of X* whose length equals that of gp. Upon
considering the elementary extremals of 7 in inverse order one sees that the
requirement that their respective lengths be the lengths (15.4) uniquely deter¬
mines these elementary extremals. Thus 7 — gp, and statement (A) is proved.
We shall now prove statement (B).
(B). The set of curves on Xjt/or which
J = c — e2,
where c is the length of gp, forms a spannable ( k — 1 )-cycle sk~ 1 belonging to gp,
'provided e is a sufficiently small positive constant.
To prove (B) we regard the portions of the (rn — l)-spheres Mq neighboring
the first points of intersection of gp with M as a set of manifolds defining a
“propcu* section” S belonging to gp in the space (ir) of p + 2 vertices. Let
(tt 0) be the point on S which determines gp. Let (i>) be a set of parameters
regularly representing S neighboring (7r0). The number of parameters (?;)
equals the index k of gp. Suppose that the set (v) — (v0) corresponds to (7 r0).
Let F(v) be the value of J(x) at the point (7 r) on S determined by (v). According
to the result of Theorem 11.2 a spannable ( k — l)-cycle associated with g can
be chosen among the cycles of broken extremals g(ir) determined by points
(71-) on S. Inasmuch as F(v) takes on a proper maximum on S at (t>) — (t>o) the
locus
F = J = c - e2
will be such a spannable (k — l)-cycle if e is a sufficiently small positive constant .
Statement (B) is accordingly proved.
We can now complete the proof of Lemma 15.1. We see that the cycle
Sk- 1 of (B) bounds below c, in fact bounds a chain of restricted curves on X*
for which
J ^ c — e2.
Thus X* is a linking A;-cycle associated with gp, and the lemma is proved.
The Corollary of Theorem 11.4 leads to the following theorem.
Theorem 15.1. The two point connectivities of the m-sphere are all zero except
the connectivities PjP(m_i), p = 0, 1, • * * , and these connectivities are 1.
This follows from the fact that the index of the extremal gp is p(m — 1).
Iiet R be a regular, analytic homeomorph of an m-sphere, and let A 1 and A 2
be any two distinct points on R . With A1 and A2 we associate a sequence of
numbers
n 0, n\y n%} • • •
248
THE BOUNDARY PROBLEM IN THE LARGE
[VII]
which we call the conjugate number sequence for A1, A2. In this sequence n*
denotes the number of extremals g (possibly infinite) on which there are k con¬
jugate points of A1 on g , including A2. From Theorem 13.3 and Theorem 15.1
we derive the following corollary.
Corollary. In the conjugate number sequence corresponding to any two distinct
points on a regular , analytic homeomorph of an m-sphere there are at most m — 2
consecutive zeros if the pair of end points A1, A2 is non-degenerate, and at most
2m — 3 consecutive zeros if the pair A1, A2 is degenerate.
When k — p(m — 1), the extremal affirmed by Theorem 13.3 to exist and have
the length Jk(A \ A2), will have at least
k — p{m — 1)
conjugate points on it including A2, while for k - (p + 1)( m — 1), the cor¬
responding extremal will have at most
(p + 1)( m ~ 1) + {m - 1)
conjugate points on it. The number of integers bet/ween these two integers is
seen to be 2 m — 3.
For example the conjugate number sequence for two points on a 3-sphere not
diametrically opposed is
1,0, 1,0, 1,0,
We are supposing the integral is the arc length. If the two points on the
3-sphere are diametrically opposed, the conjugate number sequence is
0,0, oof0, 0,0, «,0, 0,0, oo, ... ,
so that there are 2m — 3=3 consecutive zeros periodically recurring.
The preceding results lead also to the following corollary.
Corollary. On any regular , analytic homeomorph of an m-sphere there exist
infinitely many geodesics
9 i, 9 2,
joining any two fixed points A 1 and A2. The length of gn and the number of con¬
jugate points of A1 on gn become infinite with n.
CHAPTER VIII
CLOSED EXTREMALS
We continue with the Riemannian space R of the last two chapters. We shall
he concerned in this chapter with closed extremals, that is, with extremals which
return to the same point with the same direction. We shall treat the reversible
case, that is, the case in which an extremal reversed in sense is again an extremal.
The absence of end conditions of the nature of those of Ch. VII, and the
condition of reversibility necessitate a new approach to the topological aspects
of the problem. The fundamental entity here will not be the continuous closed
curve, but rather the closed broken extremal determined by a point (7 r). It
might seem at first glance that no purely topological basis could be obtained
thereby, but the contrary is the case. By considering points (x) with arbitrarily
many vertices, and by introducing an abstract semi-topological definition of
elementary extremals, we free the basic topology from the metric employed.
We shall defer this part of the theory until §12 is reached. In §12 we shall
define a metric in the small , assigning to such a metric the properties ordinarily
assigned 1o a metric, together with certain additional properties peculiar to the
needs of a theory in the large. Infinitely many metrics of the nature prescribed
turn out to be possible. The existence of such metrics as a class is a topologically
invariant fact, and the connectivities 1\, which are central in our theory, are
proved independent of the particular metric used to define them.
Our basic domain U is here the set of all spaces representing admissible points
(7 r). There are infinitely many spaces (7r) since there are points (7 r) with arbi¬
trarily many vertices. The central topological problem is the proper definition
of an homology on It is made difficult by the fact that we must regard a
circular permutation of the vertices of a point (7r) as giving an equivalent point
(7r). Another difficulty arises from the fact that the vertices of a point (71-) may
coalesce. By far the greatest difficulty, however, arises from the fact that
homologies of some sort must exist between cycles on spaces (71-) with different
numbers of vertices. To meet this difficulty we introduce special homologies
not defined in terms of bounding.
To be useful, our various conventions must lead to definitions of connectivities
which are topological invariants, and which are related to the analytic char¬
acteristics of closed extremals in the same way that the connectivities of il in the
preceding chapter are related to the type numbers of the critical sets of extremals.
In this chapter we present a solution of these problems. It is essentially a t heory
of the function space attached to closed curves on a Riemannian space R.
249
250
CLOSED EXTREMALS
[ VIII ]
The complexes Kp , and IP
1. Recall that the basic simplicial circuit K used to define the Riemannian
space R lies in a euclidean space E of g dimensions. As previously we are con¬
cerned with sets of points
(1.1) P\ • ,PP
on K. We denote such sets of points by (7 r), and represent (71-) as a point on the
p-fold product Kr of K by itself. We suppose Kv represented by a simplicial
complex in the euclidean space Ev , the p-fold product of E by itself. We shall be
concerned in this chapter with the domain of points ( rr ) on Kp in which two
points (7 r) whose vertices have the same circular order, either direct or inverse,
shall be regarded as identical. To obtain this domain one must identify the
points of Kp under transformations of a group Gp. This group we now define.
The transformations T r and Sr . We are concerned with transformations T r
of the vertices (1.1) into vertices
(1.2) Q\ ■ ■ ■ , Qr-
Under Tr the point P' is replaced by the point
(1.3) Q' = (r « 0,1, ,p - 1)
where we understand that 1 is any integer, positive, negative, or zero, and that
the superscripts in (1.3) are to be reduced, mod p , to a residue 1, • • • , p. We
are also concerned with transformations Ur under which the point Px is replaced
by the point
(1.4) Q{ = P- -•+'
with the same reduction, mod p, of the superscripts. The transformation Ur is
its own inverse. We note that Vr = U0Tr , w here wre understand that the trans¬
formation Tr is followed by the transformation U0 . The transformations Tr)
U0, and their products form a group Gp whose elements we denote by (7*.
Let the coordinates y*, h = 1, ••*,//, of the point P' in the euclidean space E
be denoted by
(1.5) vi (* = i, •■■,?).
Let us regard the set (1.5), taken for all the above values of h and z, as the co¬
ordinates of a point (t) on Ep . By an element in the space Ep we shall mean a
set of points in Ep satisfying a finite number of homogeneous linear equations
and inequalities between the coordinates of a point (tt) of Ep. We can regard a
transformation of the group Gp as a point transformation of the space Ep. Con¬
cerning Gp and Ep we shall now prove the following lemma.
Lemma 1.1. The space Ep can be divided by hyperplanes into a finite number of
elements M , with the property that under Gv an element is carried into an element ,
[1]
THE COMPLEXES K, Kp, AND IP
251
and that an element M which is carried into itself under a transformation Gi is
pointwise invariant under Gi.
Let i,j be integers of the set 1, • • * , p. Let h be one of the integers 1, • •• ,
Our elements M will be defined by subsets of the conditions
(1-6)' yl < yjt 0 ^ j),
(1.6)' yl = yl d<j).
For each pair of integers a and /3 on the range 1, • • - , p, with a < f$, the condi¬
tions (1.6) include just three conditions involving both a and fi as superscripts,
namely
Vh < yl, yl <* yh, Vh = yl-
Suppose the vertices (1.1) determine a point (x) with coordinates (1.5). Sup¬
pose that the vertices (1.1) are replaced by a new set (1.2) under a transforma¬
tion T r of the group Gv . We denote this new set of vertices by (x'). The point
(x') will possess a set of coordinates (1.5) in general numerically different from
the coordinates (1.5) of (x). If the coordinates of (x') satisfy one of the condi¬
tions A of (1.6), the coordinates of (x) will satisfy the condition B obtained from
A by replacing y\ and yl respectively by
We say that condition A corresponds to B under TV We see that T r permutes
the conditions (1.6) in a one-to-one manner preserving the equality when the
equality holds. If (x') is the image of (x) under U r we obtain the conditions
on (x) by replacing yl and yl in the conditions satisfied by (x') by
Vh{'ry y~h^r,
respectively.
Let (x) be a point of Ep and H the set of all conditions (1.6) satisfied by (x).
The set of all points on Ep which can be connected to (x) among points which
satisfy H will be termed an element H. We shall prove that the elements H can
serve as the elements M of the lemma.
Observe that two elements H are either identical or possess no points in
common. Consequently when one point of an element H is congruent to a
point of the same element, under a transformation Gi} H must be carried into
itself under Gi. It remains to prove that H is then pointwise invariant under G*.
(a). Suppose first that an element H is self-congruent under Tr , where r ^ 0,
mod p. We shall prove that H is pointwise invariant under 7V
To that end consider the set of conditions
•7) yh < yh , yh < yk , ■ • • ,
reducing the superscripts, mod p, to the range 1 , • • • , p, and holding i, h, and r
252
CLOSED EXTREMALS
[ VIII ]
fast. I say that these conditions are incompatible if r ^ 0, mod p. For if q is
the smallest positive integer such that
qr = 0 (mod p),
the gth condition in (1.7) reduces to the form
viH'~l)r < vl
and is incompatible with the ensemble of conditions preceding it in (1 .7).
Moreover no point (tt) on the element 77 can satisfy the first condition in
(1.7). For in such a case the image (x') of (w) under 7Ur would satisfy the
second condition in (1.7) by virtue of the form of T _r. But under the hypothesis
of (a), (7 r') and (rr) bot h belong to 77, so that (7r) also satisfies the second condition
in (1.7). Similarly (w) satisfies the remaining conditions in (1.7). But the
conditions (1.7) are incompatible, so that no point of 77 can satisfy the first con¬
dition in (1.7).
One can prove similarly that no point of 77 can satisfy the condition
vi > vV\
so that 77 must satisfy the equation
vi = vi*r-
Hence 7/ must be point wise invariant under Tr. The conclusion of (a) is ac¬
cordingly established.
(b). We now consider the case in which 77 is self-congruent under Ur} and
prove that 77 is then pointwise invariant under Ur.
Under Ur the regions
vi < y~k^r, yli+r < vi
are interchanged. Hence 7/ can be self-congruent under Ur only if it satisfies
the conditions
vi - *r<+r
and hence is pointwise invariant under U r.
If we take the hyperplanes (1.6)" as the hyperplanes of the lemma, the lemma
follows from the preceding analysis.
Suppose Kp is subdivided into a simplicial complex in the space Ep. Let A
be a finite set of hyperplanes of dimension one less than that of Ep, so chosen
that each 7-cell of Kp is on the 7-dimensional intersection of a subset of these
hyperplanes. Let B be the set of hyperplanes congruent under Gr to the hyper¬
planes of A. Let the cells of Kp be sectioned (Lefschetz [1], p. 67) by the respec¬
tive hyperplanes of B. Let the resulting cells be further sectioned by the set of
all hyperplanes (1.6)" The resulting complex will be divided into cells, each of
which lies on some one element of M , and is carried by transformations of
[2]
THE INFINITE SPACE tt
253
Gp into cells of the complex. So divided our complex will be again denoted
by Kp.
Let II o be the complex obtained from Kp by identifying cells of Kp which are con¬
gruent under Gp, and subdividing the resulting complex in such a manner that it is
the image of a simplicial complex.
A point (7 r) wdiose successive vertices represent points on R which can be
joined by elementary extremals on R , will be termed admissible.
The subdomain of II £ which consists of admissible points (7r) will be denoted by TP.
T*he infinite space ft
2. In this section we combine the different domains IP into a domain ft, and
introduce new conceptions of homologies necessary if the basic relations of Ch.
VII are to be preserved.
By the domain ft we mean the set of all points (tv) on the respective domains
IP. Any infinite set of fc-chains
(p = 3,4, •■•)
on the respective domains IP, all null except at most a finite number, will be
termed a k-chain z on ft. The chain zv will be termed the pth component of 2.
The sum , mod 2, of a finite number of /e-chains z on ft shall be defined as the chain
on ft w hose 7^th component is the sum, mod 2, of the />th components of the
given chains c.
A point (7 r) on IP will be termed contracted if its vertices are coincident. A
ceil on IP will be termed contracted if composed of contracted points (?r). In
determining boundaries, cycles, and homologies on IP, contracted cells shall not
be counted. With this understood the boundary of a k- chain £ on ft shall be the
(k — l)-chain whose pth component is the boundary on IP of the pth component
of z. A ft-chain on ft without boundary will be termed a A>cycle. The qualifi¬
cation mod 2 is to be understood throughout.
The relations between homologies and bounding on ft will not be the ordinary
ones by virtue of the conventions wre now' introduce. It is by virtue of these
conventions that we shall have homologies between cycles whose components
lie on different domains IP.
By the r-fold partition , r > 0, of a point (tv) on IP we mean the point (tv) on
nrp obtained by inserting r — 1 vertices on g( w) between each pair of successive
vertices Pf and P" of (tv) so as to divide the elementary extremal joining P' to P"
into r segments of equal /-length. If P' = P", the vertices added are identical
with P' and P". Let dp be a closed &-cell on IP given as the continuous image on
IP of a closed fc-simplex a. By the r-fold partition of av on IIrp, we mean the
closed cell cp oh Urp, obtained by replacing the image (r) on IP of each point of
a by its r-fold partition on ITP. The cell cv is thus the continuous image on
nrp of a. Let zp be a fc-chain on IP. By the r-fold partition of zp on IIrp we mean
the sum on nrp of the r-fold partitions of the respective closed fc-cells of zp .
254
CLOSED EXTREMALS
[ VIII ]
Let z and w be two n-cycles on 12 with pth components zp and wp respectively.
If for each p we have
(2.1) zp ~ wp (on IP)r
we shall say that we have a simple homology
(2.2) z * w (on 12).
On the other hand let z be a &-cycle on 12 writh at most one non-null component
zv . We shall refer to z as the cycle zp on 12. In f he same sense let w Q be a second
fc-cycle w on 12. If wq is the r-fold partition of zp we shall say that we have a
special homology
(2.3) z * w (on 12).
We shall also write
(2.4) zp * wq (on 12).
We now define an 12-homology as one formally generated by the addition, mod 2,
of the respective right and left members of a finite set of simple and special
homologies between A:-cycles on 12. If the resulting sums are cycles u and v
respectively, we write
(2.5) v * v (on 12).
We also write (2.5) in the form u + v * 0. Our 12-homologies thus admit
the usual formal linear operations.
We note that a set of generating homologies sum to a homology of the form
(2.5) in which u and v are unique fc-cycles on 12. On the other hand an
12-homology of the form (2.5) can be generated by the addition of simple and
special homologies in infinitely many ways. Unlike an ordinary homology an
12-homology (2.5) does not in general imply that its members bound a chain on
12. A simple homology in a set of generating homologies does howrever imply
that its members bound a chain T on 12. We term such a chain T a hounded chain
implied by the corresponding homology. An 12-homology will be said to hold
on a subdomain 120 of 12 if the members of a set of generating homologies are on
120 and if the members of the respective simple homologies bound chains on 120.
Suppose wre have an 12-homology of the form
(2.6) ^ * 0.
Let H be a set of simple and special homologies generating (2.6). The simple
homologies of H can be combined into homologies of the form
(2.7) up — 0 (on IP),
the homologies (2.7) including just one homology for each integer p, with all
[2]
THE INFINITE SPACE a
255
but a finite set of the cycles up null. Suppose there are n special homologies in
the set IL The zth of these special homologies will then take the form
(2.8) vPi * wQi (i = 1, • • • , n)
where the two members of (2.8) are A;-cycles on TIPi and n9» respectively, and one
of these cycles is the partition of the other with ^ qt. The pth component
of z in (2.6) will then be a fc-cycle zv on IF of the form
(2.9) = wp + + 5qpiwqi (i = 1, ■ • ■ , M),
where 8] is the Kronecker delta, and the terms involving i are to be summed with
respect to i, holding p fast.
A chain on II p will be said to have an index p. A chain :onll will be said to
possess an index equal to the least common multiple of the indices of its non-null
components. A set of homologies generating an Q-homology will be said to
possess an index equal to any multiple of the members of the respective generat¬
ing homologies. If an O-homology 2*0 can be generated by a set of homologies
with index p} we shall say that z * 0 with index p.
Let 2 be a chain on 0 with index p. Let q be any positive integral multiple
of p. If each non-null component of 2 is replaced by its partition on H9, and the
resulting chains added mod 2, on n9, one obtains a chain wq on n9 which will be
termed the partition of 2 on n9. With this understood we state the following.
A ny k-cycle 2 on U such that 2*0, possesses a partition wq ~ 0 on II9, provided q
is a suitably chosen positive integer.
Let q be an index of a set of homologies generating the homology 2*0. Each
of these generating homologies implies a homology on II9 between the partitions
on II9 of the members of the given generating homology. In particular a special
homology thereby implies an homology on n9 between identical cycles. If
wq is the partition of 2 on n9, it appears from the definition of an O-homology,
that the homologies which we have obtained on n9 sum to an homology reducible
to the form
w 9 0 (on n9).
The statement in italics is accordingly proved.
A set of A>cycles on 0 will be termed Sl-dependent if a proper linear combination
of cycles of the set is 12-homologous to zero. By the connectivity
Pk (k = 0, 1, 2, •■•)
of 12 we mean the number of fc-cycles on 12 in a maximal set of 12-independent
fc-cycles. We admit that Pk may be infinite. The following statement covers a
particular case of interest.
A necessary and sufficient condition that the connectivity P0 of 12 be null , is that
every closed curve on R be deformable into a point on R.
We shall prove the condition sufficient. To that end let (t) be any point on
IL We regard (71-) as a 0-cycle on 12, and wish to show that (71-) * 0 on !L The
256
CLOSED EXTREMALS
[ VIII I
curve g(ir) is deformable into a point on R by hypothesis. One sees then that a
sufficiently high partition of (t) on IP can be deformed on IP into a contracted
point (to) on IP. Hence
(t) * (tt0) (on 12).
But according to our conventions the contracted point (to) can be omitted in the
count of boundaries so that (t) * 0 on 12.
Conversely it will follow from our latter work that the condition is necessary.
Inasmuch as we shall not use this fact we omit further details.
Critical sets of extremals
3. In this section we shall make a study of the existence of closed extremals
from the point of view of the theory of analytic functions. To that end we shall
say that a continuous family of closed curves is connected if any closed curve of
the family can be continuously deformed into any other closed curve of the
family through the mediation of curves of the family. A connected family of
closed extremals which is a proper subset of no connected family of closed ex¬
tremals will be called a maximal connected set of closed extremals. We shall
prove the following theorem.
Theorem 3.1. The number of maximal connected sets of closed extremals on which
J is less than a constant b is finite. On each such set J is constant.
Let g be a closed extremal of /-length a>. Let g be given a positive sense.
Let Q be a point on g. The neighborhood of Q can be represented regularly and
analytically in terms of coordinates
O, yi, ■■■ , vJ
such that along g neighboring Q
Vi = * • • - yn = 0 (n = m - 1)
and x is the arc length on g measured from Q. The extremals neighboring g
can be represented near Q by giving the coordinates yi of their points as functions
<pt(x, a) of x and 2 n parameters (a) which give the initial values of ( y ) and (yf)
when x = 0. The functions < px{xt a) will be analytic in their arguments for x
near 0 and (a) near (0).
For sets (a) sufficiently near zero the extremal ga , determined by (a) when
x = 0, will return to the neighborhood of Q after traversing a /-length o>, and
will then be representable in the form
Vi = a),
where ^i(xy a) is analytic in x and (a), for (xy a) near the set (0, 0). In order
13]
CRITICAL SETS OF EXTREMALS
257
that ga be periodic with respect to its J-length, and possess a period near w, it is
necessary and sufficient that
(Pi( 0, a) = MO, <x)y
(3.1)
<1 Ptx(0 , a) = ^ti(0, a),
for (a) sufficiently near (0).
The equations (3.1) may be satisfied for real sets (a) only when (a) = (0), or
they may be satisfied identically. Apart from these special cases the real solu¬
tions (a) of (3.1) will be representable as functions “in general” analytic on one
or more locally connected “Gebilde” G (Osgood [1]) of r independent variables
with 0 < r < 2 n. Each G includes the point (a) — (0). To each set ( a ) on G
corresponds a closed extremal. Corresponding to any regular curve T on one
of the above Gebilde G, one obtains a 1-parameter family of closed extremals.
Upon differentiating the J- lengths J of these closed extremals with respect to
the arc length along T we see that J' = 0. It follows that J is constant on G,
in fact takes on the value a>.
To come to the theorem let us suppose' the theorem is false in that there exist
infinitely many maximal connected sets of closed extremals on which J < b.
In each of these sets we choose an extremal g*, and on g* a point p. Let X be a
unit contravariant vector tangent to g* at p. Let L be the J-length of g *.
The sets (p, X, L) are infinite in number. They have at least one cluster set
(p0, X0, L0). We see that 0 < L0 ^ b.
The extremal go passing through the point p0 with the direction X0 will be
closed, and with respect to its J-length possess a period Lo. But as we have
seen in an earlier paragraph, closed extremals sufficiently near g0, with periods
sufficiently near Lo, will be connected to g0 among closed extremals of the same
class, contrary to the choice of (po, X0, L0) as a cluster set of the sets (p, X, L).
From this contradiction we infer the truth of the theorem.
Corresponding to any point (7 r) on IB let the value of J taken along g(rr) be
denoted by J(7r). Neighboring any point (7r0) we can regard J(7r) as a function
\J/ of the coordinates (x) in the sets locally representing the vertices of (x) on R.
The function \p will be analytic provided consecutive vertices of (t) remain
distinct. Of the points (tt) whose successive vertices are distinct, a point (71-)
at which \(/ has a critical point will be called a critical point of J( tt). As in Ch.
VII, so here, it follows that a necessary and sufficient condition that a point on
IF with consecutive vertices distinct be a critical point of J( 7r), is that g(ir)
be a closed extremal. We define a critical set of J(tt) on IF as a set of critical
points of J(7r) on which J( w) is constant, and which is at a positive distance
from other critical points of
By a critical set of closed extremals we mean a set of closed extremals on which J
is constant, and which contains the whole of each maximal connected set of
closed extremals of which it contains a single extremal. A critical set of closed
extremals will be termed complete if it contains all of the closed extremals on
which J equals a given constant c.
258
CLOSED EXTREMALS
[VIII]
Let A be a critical set of closed extremals of /-length c. In terms of the
constant p of Ch. VII §2, let p be an integer so large that pp > c. Of points
(7r) on IIP whose consecutive vertices are distinct let ap be the set which deter¬
mines extremals g( 7r) of A. We see that ap is a critical set of /( ir) on IF.
The ensemble of the sets
‘ ' •
will be termed the critical set a on 12 determined by A. The set <rp will be termed the
component of a on IF.
The domain IF
4. In this section we shall begin an analysis of the domain IF, and the critical
sets on this domain. This analysis is preliminary to a similar analysis of the
domain 12. In the present section p is a fixed integer greater than 2.
A first difference between the developments in the present section and those
of Ch. VII arises from the convention that contracted ( Jc — l)-cells are omitted
from the boundaries of /c-chains on IF. A necessary and sufficient condition
that a point (w) be contracted is that as a point of Kp it be invariant under the
transformation Tl of §1. Now the cells of Kp have been so chosen as to be
pointwise invariant under T i whenever they possess a pair of points congruent
under Tx. Hence if one point of a cell of IF is contracted so are all points of that
cell.
We shall now prove the following lemma.
Lemma 4.1. Corresponding to an arbitrary positive constant b there exists an
arbitrarily small positive constant 8(b), such that any k-cycle on IF below 8(b) is
homologous to zero below b on IF.
A cycle zp on IF below a sufficiently small positive constant will be arbitrarily
near the subcomplex of contracted cells of IF, and will be homologous to a cycle
up of contracted cells. Moreover there exists a chain on IF bounded by zv
and up arbitrarily near the contracted cells of IF, if zp itself is sufficiently near
these contracted cells. This follows readily from the Veblen- Alexander defor¬
mation. By virtue of our conventions the cycle up of contracted cells can be
dropped from the homology zp ~ up , so that zp ~ 0.
The statement of the lemma involving b and 8(b) follows from the fact that
J(n) is continuous on IF, and equals zero on the contracted cells.
A second departure from Ch. VII comes in new demands which we must put
on “/-deformations” of points (ir) on IF. As in Ch. VII such deformations
should carry admissible points (w) into admissible points (ir) and not increase the
value of /( 7r) beyond its initial value. But in the present chapter points (t)
on Kp which are obtained one from the other under transformations of the group
Gp must be deformed through points with the same property. Moreover our
deformations should vary contracted points through contracted points.
In §3, Ch. VII, we made use of a deformation DffDf . We shall now define
[4]
THE DOMAIN IP
259
deformations D* and Z)**, analogous to D' and D" respectively. The deforma¬
tion D* tends to distribute the vertices of a point (x) more evenly on g(ir)f while
D** tends to decrease the ./-length of /( tt) if the vertices of g(-w) are already
fairly evenly distributed on g( tt).
The deformation D*. Let (7r) be a point on IP. Let g* be an unending curve
“covering” g(ir). On g * let s represent the /-length measured in a prescribed
sense from a prescribed point on g*. The vertices of (7r) will be represented by
infinitely many copies on g *. Let
(4.1) P\ • • • , P*
be a set of copies of consecutive vertices of (tt) which appear consecutively on
g* in the order (4.1). If c is the /-length of g( tt) let
(4.2) Q\ * * - ,
be a set of consecutive points on g* which delimit successive segments of g* of
/-length c/p , and which are so placed that the average value of s for the points
(4.2) is the same as for the points (4.1). The deformation D* is now defined as
one in which the vertices (4.1) move along g* to the corresponding vertices in
(4.2) , moving at /-rates equal to the /-lengths to be traversed on g*.
One sees that the point (w') on IP which is determined by the vertices (4.2)
will be independent of the particular set (4.1) chosen as above to represent the
point (7 r). In particular one might replace P1 in (4.1) by a point on g* for which
,s is c greater. The average s for the points (4.1) is now c/p greater than pre¬
viously. The corresponding new set (4.2) will now be obtained from the old set
(4.2) by replacing Q 1 by a point on <7* for which s is c greater. But this new
set (4.2) determines the same point (7 r') on IP. One also sees that the point
(7 r') determined by (7 r) is independent of the sense assigned to g* and of the point
from which s is measured. The same is true of the points through which (71-)
is deformed under D*. Thus D* has the properties required of a /-deformation.
The deformation D**. We begin by assigning a metric to IP neighboring
points (t) whose consecutive vertices are distinct. Let (7 r0) be such a point and
Pl0t ••• ,PJ
its successive vertices. Let
(4.3) x\, ,xmq {q = 1, ,p)
be local coordinates on R neighboring PJ- Suppose that the form
ds\ = g^dx^dzl (q not summed)
defines the metric on R neighboring the vertex PJ. We then assign the metric
(1 q summed)
(4.4)
ds2 = g^jdXqdx^
(iy 3 = 1, • • • , m;q = 1, * • • , p)
260
CLOSED EXTREMALS
[ VIII ]
to IIP neighboring (7 r0). We note that the form (4.4) is invariant under the
transformations of points (7 r) on Kp which are defined by members of the group
Gp, so that the form (4.4) is uniquely defined on IF.
In conformity with Oh. VII we denote the set of all parameters (4.3) by (u),
and write (4.4) in the form
(4.5) ds2 — b hk(u)duhduk (h, k = 1, ■ • • , nip).
We denote the value of J(t) at the point determined by ( u ) by <p(u).
Let b be any ordinary value of J . Suppose moreover that pp > b > 5(b)
where 5(b) is the constant of Lemma 4.1. Suppose further that 5(b) is less than
any critical value of J. Let 2 denote the set of J-normal points of IF for w hich
(4.6) 5(b) S J( tt) ^ b .
Let 7j be a positive constant so small that any point on IF within a distance 7/
on IF of points of 2 will define elementary extremals of positive ./-lengths less
than p. Recall that it is only for the case of distinct consecutive vertices that
<p(u) is assuredly analytic.
We now define D** in the same manner as D" was defined in Ch. VII, making
use of the preceding metric (4.5) of 2, and of the preceding constant 77. The
deformation ])** is uniquely defined for each point of IF, by virtue of the in¬
variance of the form (4.4) under transformations of the group (P\ and in partic¬
ular by virtue of the corresponding invariance of the trajectories (3.6) of Oh. VII.
Let Ri denote the connectivities of the domain J (tv) < b on IF. We take these
connectivities in the ordinary sense, modified only by our conventions con¬
cerning the omission of contracted cells. We shall prove the following theorem.
Theorem 4.1. Let a and b, a < b, be ordinary values of J between which there
are no critical values of J . The connectivities of the domains J (tv) < b and J (tv) < a
on IF are finite and equal . If there are no critical values less than b , the connec¬
tivities of the domain J(tv) < b are null .
We begin with a definition.
The deformation I) *. We replace the deformation Dp = D"Df of §3, Ch.
VII, by the deformation
D* - D**D*.
With its aid wre shall now prove statement (a).
(a). The connectivities of the domain
(4.7) J(tt) < b (on IF)
zre finite.
As in Ch. VII so here it follows that D* will deform the domain (4.7) into a sub-
domain H whose boundary points are inner points of the domain (4.7). If
IF is sufficiently finely subdivided, a subcomplex T of IF can then be chosen so
as to contain all the points of //, and to consist of points of (4.7). Any fc-cycle
[5]
CRITICAL SETS ON IP
261
on IT is thus deformable on IF under Z>* into a fc-cycle on T. Moreover, D*
deforms a cycle through cycles, since Z)* deforms contracted cells through
contracted cells.
To show, as in Ch. VII, that any fc-cycle 2 on (4.7) is homologous (not counting
contracted cells on boundaries) to a fc-cycle of cells of T we use the Veblen-Alex-
ander process. In this deformation a point of 2 never leaves the closed cell of T
on which it is originally found, and this has the consequence that contracted cells
are deformed through contracted cells. Hence 2 will be deformed through
fc-cycles into a fc-cycle. The connectivities of (4.7) will then be at most the con¬
nectivities of T and will thus be finite. Statement (a) is accordingly proved.
To prove that the connectivities of the domain (4.7) are null if there are no
critical values less than b we note that a sufficient number of iterations of D * will
deform a fc-cycle on (4.7) into a fc-cycle on the domain
(4.8) J(tt) ^ 5(b) ,
where 8(b) is the constant of (4.6) and Lemma 4.1. But by virtue of Lemma 4.1
all cycles on (4.8) are homologous to zero on the domain J(r) < b. Hence the
connectivities of the domain (4.7) are null if there are no critical values less
than b.
Finally the connectivities of the domains
(4.9) J (7 r) < a, J(ir) < b
are equal if there are no critical values of J between a and b. This follows irom
the result of the preceding paragraph if both a and b are less than the least critical
value of J (t). In case a and b are both greater than the least critical value of
J(tt) the connectivities of the domains (4.9) are again equal as can be proved
formally after the manner of proof of Theorem 3.2, Ch. VII.
The proof of the theorem is now complete.
Critical sets on IF
5. Let f be a critical set of closed extremals on which J = c. Suppose the
integer p so chosen that pp > c. Let ap denote the set of critical points on IF
which determine curves g(ir) of f. We state the following theorem.
Theorem 5.1. The set 2 of all j -normal points on IF neighboring crp forms a
regular analytic Riemannian manifold.
The proof of this theorem can be brought under the proof of Theorem 7.1,
Ch. VII, as follows.
Let (tt0) be any J-normal point of <rp. Let the vertices of (7r0) be denoted by
P1 . . . pp
* 0 j > * o>
taking these vertices in one of their circular orders. Let (7r) be a J-normal point
262
CLOSED EXTREMALS
[ VIII ]
neighboring (7r0). Let the vertices of (tt) neighboring the respective vertices of
(7r0) be denoted by
p\ - , rp-
To make use of the proof in Ch. VII we cut g(ir) at Pp , forming thereby a broken
extrema] whose end points A1 and A2 both cover Pp, but are regarded as distinct.
In the notation of Ch. VII this broken extremal possesses the vertices
A\ P1, • , Pp-\ A2,
where A1 and A2 are subject to end conditions which make A 1 and A2 cover the
same point on R. The end parameters of Ch. VII may be taken as any set of
admissible coordinates of R representing A1 or A 2 neighboring P%. One now
completes the proof as in Ch. VII.
We continue with the following theorem.
Theorem 5.2. The value of J(tt) on the subspace 2 of J-normal points (t)
sufficiently near <rp is an analytic function of the local coordinates ofHy and possesses
no critical points other than J-normal points of the set op.
The proof of this theorem does not differ essentially from the proof of Theorem
7.2, Ch. VII, and will be omitted.
The radial deformation RP(t), 0 :§ t < 1. Let <p be a neighborhood function
belonging to the function- J( tt) on 2 and to the critical set of that function.
Let r be a positive constant so small, that the points on 2 which are connected to
ap, and for which
*P = r>
form a closed domain at each point of which p> enjoys the properties of a neigh¬
borhood function. With the aid of<?we introduce “radial trajectories” on 2
neighboring ap as in Ch. VI. Under the deformation Rp(t ) a ./-normal point P
at which
ip — r — dr (0 ^ 0 < 1)
shall remain fixed until t reaches the constant 0, and shall thereafter be replaced
at the time t by the point on the radial trajectory through P at which <p — r — tr.
The deformation Rp(t) is thereby defined.
We continue with the following lemma, the analogue of Lemma 8.1 of Ch. VII.
Lemma 5.1. There exists a J -deformation Ep(t ), with time interval 0 ^ t ^ 2,
which deforms points ( ir ) on IP neighboring <rp into J-normal points (w)y and
leaves J-normal points invariant. Moreover Ep{t) deforms the vertices of each point
(ir) on <rp through points (tt) on the extremal g( tt) into a J-normal point of cp.
We begin with the deformation D* of §4, thereby deforming points {tt) on IP
sufficiently near ap into points (t) arbitrarily near J-normal points of ap. We
then continue with a deformation A* defined as follows.
[5]
CRITICAL SETS ON W
203
The deformation A*, 0 ^ t ^ 1. This deformation is a deformation of points
(t) on IF formally defined as was D*, except that the points (4.2) shall here be a
set of consecutive points on g* in §4 which define a J-normal point (7r') with
vertices on g( 7r), and, as in §4, are so placed that the average value of $ for the
points (4.2) is the same as for the points (4.1). If the given point (tt) is suffi¬
ciently near a J-normal point of <rp, we see that (tt') is uniquely determined,
lies on IF, and varies continuously with (tt).
The deformation Ep(t), 0 ^ t ^ 2. We now set
Ep(t) = A */>*,
understanding that D* occupies the first unit interval of time in Ep(t)y and A*
the second unit interval of time. We see that Ep(t) so defined satisfies the
requirements of the lemma.
The deformation 6p(t), 0 ^ t < 3. Proceeding formally as in Ch. VII, we now
combine the deformations Ep(t) and Rp(t) of this section into the deformation
d,,(t). We understand that in Bv(t)y Ep(1) occupies the time interval 0 tk t ^ 2
and Rp (t) the time interval 2 rg t < 3. The deformation 6p(t) is applicable to
points ^7r) sufficiently near ap. Its characteristic properties are enumerated as
follows-
Theorem 5.3. Under 0p(t)y 0 ^ t < 3, ap in deformed o?i itself. For t g 2
each point of IF which is sufficiently near ap is replaced by a J -normal point (tv).
Any sufficiently small neighborhood of ap is deformed at the time t into a neighbor¬
hood Npt1 the superior limit of the distances of whose points from ap approaches
zero as t approaches 3. Under 0p(t) points below c are deformed through points
below c.
Let a and b be two constants which are not crit ical values of Jf and between
which c is the only critical value of J . Let the integer p be so chosen that pp > b .
Denote the complete set of critical points on IF corresponding to the critical
value c by ap. In terms of the deformation I) * of §4, we state the following
lemma.
Lemma 5.2. Let A:p be an arbitrary neighborhood of op on IF , and let Lv be the
set of points on IF below c . A sufficient number of iterations of the deformation
D * will afford a deformation Ap which will deform the domain J(tv) < b on itself
onto Np 4“ LP .
If a k-cycle z lies on a domain Np + Lp for which Np is a sufficiently small
neighborhood, of ap, and if z ^ 0 on J (tv) < b ( below c)f then z ^ 0 on the domain
Np + Lp ( below c).
This lemma is the analogue of the Deformation Lemma of §6, Ch. VI. Its
proof is the counterpart of the corresponding proof in Ch. VI, J replacing /,
and Z>* replacing D.
Let N*p be a fixed neighborhood of ap whose closure is interior to the domain on
which the preceding deformation 6p(t) is defined .
264
CLOSED EXTREMALS
[VIII]
Certain lemmas and theorems of Ch. VI now hold here in the same form as in
Ch. VI, except at most for the substitution of J( t) for/, 0p(t) for 6(t ), and ap for <ry
and the addition of the superscript p to the neighborhoods. The basic domain
is IF. We shall add a star to a theorem of Ch. VI to indicate that it shall be
taken with the present interpretations.
We first take over Corollary 3.1 of Ch. VI, denoting its counterpart here by
Corollary 3.1*. With the aid of the neighborhoods N*p and Mr(X) appearing
in Corollary 3.1*, admissible pairs of neighborhoods VPWP of <xp are formally
defined as are the neighborhoods VW of Ch. VI. We then add the definition of
spannable and critical k- cycles corr VpWpy belonging to <jp, as before. We
next have Theorem 5.2* where the domains <p ^ e and <p — e are to be inter¬
preted as the domains of J-normal points (7r) of the present section. From
Theorem 5.2* we infer that the number of cycles in maximal sets of spannable
and critical fc-cycles corr YPWP is finite.
Linking and invariant fc-cycles on IF corr VPWP, are now formally defined as
in Ch. VI, §6. We then obtain Lemmas 6.1*, 6.2*, 6.3* and 6.4* from the cor¬
responding lemmas in Ch. VI. The proofs are unchanged except in notation
and connotation. In proving Lemma 6.4* one replaces the deformation A (t)
of Ch. VI by a deformation A p(t) defined as follows.
The deformation A p(t)y 0 g t < 3. The deformation Ap(2) is defined in terms
of 0p(t ), as A(t) was defined in terms of 6(t) in Ch. VI. It is defined for all
points (tt) on the domain J(tt) < 6, and is continuous on this domain. It is
identical with the deformation Bp{t) of Theorem 5.3 on the neighborhood V*p of
that theorem.
Lemma 5.2 of the present section is used in place of the Deformation Lemma
of Ch. VI. With its aid one proves Theorem 6.1*. We restate Theorem 6.1*
as follows.
Theorem 5.4. A maximal set of k-cycles on the domain J < b of IF, independent
on J < by is afforded by maximal sets of criticaly linking , and invariant k-cycle e
corresponding to an admissible pair of neighborhoods VpWr of the critical set ap.
Theorem 5.3, characterizing Bp(t)y and Theorem 5.4, together with the defor¬
mation Ap(£), will be frequently used in the sequel.
Critical sets on ft
6. Let 0- be a critical set on 12 on which J = c. Let ap be the component of a
on IF. Corresponding to those integers p for which pp > c let Np be an arbitrary
neighborhood of <rp on IF. When pp S c we shall understand that Np is null.
The set of neighborhoods
(6.1) N\ N*y • • •
will be termed a neighborhood N of a on 12. The neighborhood Np in (6.1) will be
termed the component of N of index p. A neighborhood N of a will be termed
arbitrarily small if the components Np of N can be taken as arbitrarily small
[6]
CRITICAL SETS ON ft
265
neighborhoods of their respective sets ap. If X and N are two neighborhoods
of a such that
(6.2) XpCZN p (p = 3,4, - ),
we shall write X C N.
Let N* designate the neighborhood of a on 12 whose component on IF for
pp > c is the neighborhood N*p of crp} defined in §5. Let X be an arbitrary
neighborhood of a on 12 such that
(6.3) X C N *.
Corresponding to X, a neighborhood M{ X) of a on 12 will be chosen with the
following properly.
(A). Let p be any positive integer , and q any integral multiple of p , The com¬
ponent M9(X) shall be so small that any point (t r) which lies on MP(X), and pos¬
sesses a partition on M9(X)} will be deformed under 8p(t) through points which
possess partitions on X9.
We do not exclude the case where p — q. When p = q the preceding condition
means that Mp( X) shall be so small that any point (t) on Mp( X) will be de¬
formed under 0p(t) on Xp. This part of the condition on MP(X) is similar to the
condition on MV(X) of §5.
It is clear that the neighborhoods Mq(X) can be successively chosen in the
order of the integers q so as to satisfy the preceding conditions. With this
understood we state an analogue of Corollary 3.1 of Ch. VI. Entirely new con¬
siderations enter into its proof.
Theorem 6.1. If X is an arbitrary neighborhood of o on N*y any k-cycle z on
M (Ar) ( below c) is U-homologous on X ( below c) to a k-cycle ( below c) on an arbi¬
trarily small neighborhood N of a.
Corresponding to an arbitrarily small neighborhood N of a there exists a
neighborhood No of a whose components are so small that any k-cycle z on No} such
that z * 0 on M(N*) ( below c) uwith index ” q , will have the property that z * 0 on N
( below c) with index q.
To see that the k- cycle z on A/(Ar) is 12-homologous on A" to a cycle on N we
have merely to apply the deformations 6p(t) to the corresponding components
zp of zy continuing Bp{t) up to a suitable time /, 0 ^ K 3, dependent on Np .
The cycle zp will thereby be homologous (below c) to a cycle on Np (below c).
We shall now prove the second statement of the theorem.
To that end let A"0 be a neighborhood of a for which Nq is so small that it is
deformed only on Np under 8p(t). For this choice of No we shall prove that the
second statement of the theorem is true.
We suppose then that z is a fc-cycle on N 0 such that
(6.4)
2*0
[on M(N*)].
266
CLOSED EXTREMALS
[ VIII J
As we have seen in §2 the S2-homology (6.4) can be obtained by the formal
addition of a set of ordinary homologies
(6.5) up ~ 0 [on M(N*), p = 3, 4, •••]
and a set of /x special homologies of the form
(6 6) vp * * wQi [on M(N*), i = 1, • • • , m]-
As we have noted, all but a finite set of the cycles uv are null. Moreover, as in
§2, we have
(6.7) zp — up bppH)Pi + Spu**' (i = 1, • * * , m),
summing with respect to z.
Suppose that p, > q, in (6.6). The cycle vPi is then a partition of the cycle
wq\ By virtue of the conditions (A) on M(N*), the k- cycle wQi can be deformed
under 0qi(t) on N*9i into a fc-cycle arbitrarily near crQi in such a fashion that
the partition of wqi remains on N*Pi. We accordingly infer the existence of
homologies
wq' ~ wq< (on N*q*)t
(6.8)
vpi ~ vpi (on N*Pi ),
in which the fc-cycle vp* is a partition of wq', and the cycles vPi and wq' lie on N0.
We record the special homologies
(6.9) vp0< * wqQi (on N o,i = 1 , • ■ • , n).
If we set
(6.10) ul = up + 5p*(vPi - vPi) + 6qi(wqi - wqj) (i = 1, • • • , n),
summing with respect to i, we see that (6.7) can be given the form
(6.11) zpze up + 8pp*vp0* + Spur!*.
From (6.11) it appears that up lies on Np , since the remaining cycles in (6.11) lie
on Np. From (6.5), (6.8) and (6.10) we see that
(6.12) ul ~ 0 (on Ar*p).
But from (6.11) it follows that the homology (6.4) may be regarded as generated
by the homologies (6.12) and the special homologies (6.9). Observe that the
cycles in these generating homologies all lie on No, and that an index of the
original homologies (6.5) and (6.6) is also an index of the homologies (6.9) and
(6.12) .
It remains to show that the homologies (6.12) imply homologies
(6.12)' u%~0
(on Np).
[6]
CRITICAL SETS ON Si
267
To that end let vp denote the chain on N*p bounded by Uq. The deformation
6p{t) applied to vp up to a suitable time to, will deform vv into a chain xp as near
<rp as we please, in particular into a chain on Np. But this same deformation
Bp(t) will deform u\, according to the choice of No, through a cycle wp on Np.
Hence
xp -f- wp — ► up (on Np)f
and (6.12) ' is established. The proof of the theorem is now complete.
By an admissible pair of neighborhoods VW of a we mean neighborhoods such
that
rcl(F), WCZM(V).
We shall understand that a given pair of neighborhoods VW is admissible unless
otherwise stated.
By a spannable k-cycle, corr YW , we shall mean a k-cycle on W, below cy bound¬
ing on W but not U-homologous to zero on V below c.
By a critical k-cycle, corr VW, we shall mean a k-cycle on W, not il-homologous
on V to a k-cycle on V below c.
We call attention to the fact that the distinction between bounding relations
and 12-homologies makes a real difference in the above definition of spannable
fc-cycles. We shall prove in §9 that the number of /c-cycles in maximal sets of
critical and spannable A>cycles, corr VW, is finite. We shall use this fact in the
remainder of this section.
Theorem 6.1 leads readily to the following theorem.
Theorem 6.2. Corresponding to two admissible pairs of neighborhoods VW and
V'W' of a, there exist common maximal sets of spannable and critical k-cycles on
an arbitrarily small neighborhood of a.
By a linkable k- cycle corr VW we mean a spannable fc- cycle, corr VW, which
bounds a chain on 12 below e. The present theory here departs from the earlier
theory in that in §5 a linkable k-cycle corr VpWr could be defined either as
above, or as a spannable k- cycle, corr VVWV , homologous to zero below c. In
§5 these two definitions would have been equivalent. In the present theory
however we cannot replace the condition of bounding below c, by the condition
of being ^-homologous to zero below c.
Let l be a linkable (k — l)-cycle corr VW. We now formally define a A>cycle
X linking l, as in Ch. VI. The components of X and l on IF are denoted by Xp
and lp respectively. I say that there is at least one integer p for which Xp links
lp on IF corr VPWV, in the sense of §5. For Xp could fail to be linking in this
sense only if lp bounded on Vp below c . If, for each integer p, lp bounded on Vp
below c, the cycle l would bound on V below c, contrary to hypothesis. Thus a
linking cycle X has at least one component Xp which is linking corr VPWP on IF
in the sense of §5.
On the other hand a cycle lp on IF which is linkable corr VPWP, in the sense of
268 CLOSED EXTREMALS [ VIII ]
§5, need not be linkable on 0 corr VW, since lp may be ^-homologous to zero on
V below c, and hence not be spannable corr VW.
An invariant k- cycle corr VW will be defined as a k- cycle on U below c,
^-independent below c of spannable ^-cycles corr VW. Invariant linking, and
critical ^-cycles corr VW, on i2, will be distinguished from cycles which are
invariant, linking, and critical A;-cycles on IP, corr VPWP, in the sense of §5, by
the use of the qualifying phrase corr VW, instead of the phrase corr VPWP , used
in §5.
With this understood we now establish a basic lemma by means of which the
invariant, linking, and critical cycles on IP, corr VPWP, can be expressed in terms
of maximal sets of cycles, corr VW. In this lemma we shall refer to the domain
on Q which consists of points (t r) below c as the domain L. The lemma follows.
Lemma 6.1. (a). A k-cycle on IP which is an invariant k-cycle corr VPWP, is
U-homologous on V + L to a linear combination of invariant k-cycles corr VW.
(b) . A k-cycle on IP which is a critical k-cycle corr VPWP is il-homologous on
V + L to a linear combination of invariant and critical k-cycles corr VW.
(c) . A k-cycle on IP which is a linking k-cycle corr VPWP, is U-homologous on
V + L to a linear combination of linking , critical and invariant k-cycles corr VW.
Statement (a) is true of an invariant k- cycle corr VPWP, because it is true more
generally of any A;-cycle on U below c. This follows at once from the definition
of an invariant A;-cycle corr VW.
Statement (b) is true of a critical k- cycle corr VPWP, because it is true more
generally of any fc-cycle on W. This follows from the definition of a critical
k- cycle corr VW.
We come therefore to the proof of statement (c). We suppose zv is a A>cycle
on IP which is a linking cycle corr VPWP. The k- cycle zp links a (k — l)-cycle
up , corr VPWP , by hypothesis. If up is not ^-homologous to zero on V below c,
zp is a linking fc-cycle corr VW, by virtue of the definition of such cycles, and
statement (c) is true.
It remains to prove that statement (c) is true when
(6.13) up * 0 (on V, below c).
Suppose q is an index of the homology (6.13), that is a multiple of the indices
involved in a set of homologies generating (6.13). Let N be a neighborhood of <r
which is so small that when r is a divisor of q, points (ir) on Nr have partitions on
Wq. Corresponding to N let A^o be a neighborhood of a chosen as in Theorem 6. 1 .
Let the cycle zp be deformed under A p(t) into a cycle z\ on No + L. Suppose
that up is thereby deformed into a (k — l)-cycle up0 . Since zp lies on Wv + Lp ,
and up lies on Wp below c, these deformations imply the respective homologies
Zp ^ Zq
(6.14)
(6.15)
up ~ up
(on V + L),
(on V, below c).
[6]
CRITICAL SETS ON fl
269
From (6.13) and (6.15) we see that
(6.16) 0 (on V, below c).
According to our choice of No, (6.16) implies an 12-homology
(6.17) Mq * 0 (on N, below c).
By virtue of the choice of N, zrQ and upQ possess partitions z\ and on
W9 + Lqt as do the chains involved in the homologies generating (6.17).
Hence (6.17) implies that
(6.18) u9 ~ 0 (on W9f below c).
But z\ is the sum of a A>chain on Wqy and a chain below c, with u9 as the
common boundary. It follows from (6.18), that for suitable integers ra, n,
(6.19) z\ ~ me9 + neQ (on V9 + Lq)
where cq is a critical /r-cycle corr VVW9, and eq is a fc-cycle on IIfl below c. The
special homology
2? * z\ (on V + L)
and the homologies (6.19) and (6.14) combine into the 12-homology
zp * mcq + neq (on V + L).
The proof of the lemma is now complete.
Let maximal sets of linking, critical, and invariant ^-cycles corr VW be repre¬
sented by
(6.20) (X)a-, (c)kf (i)k9
respectively. It will follow from the results of the next two sections that the
number of cycles in these sets is finite. Let q be a multiple of the indices of these
cycles. Let Ar C W be a neighborhood of a which is so small that for integers p
wdiich are divisors of q the components Nv of N possess partitions on W9. Let
each non-null component of the cycles (6.20) be deformed under the deforma¬
tions A p(t) of the same index, into a cycle on N + L. The sets (6.20) will thereby
be replaced by sets
(6.21) (A)*, (c)*, (t)*,
which remain maximal sets of linking, critical, and invariant k- cycles corr VW.
Finally let each cycle in the sets (6.21) be replaced by its partition on n5, and
the resulting maximal sets be denoted by
(6.22) Wk> (c)ky W* fc-
The resulting cycles will be 12-homologous to the cycles which they replace and
will again constitute maximal sets of linking, critical, and invariant ^-cycles
270
CLOSED EXTREMALS
[ VIII ]
corr VW . These final cycles will each possess but one component which is not
null, namely a component on II9.
Our second lemma is the following.
Lemma 6.2. There exist maximal sets of linking , critical , and invariant k-cycles
corr VW, all of whose components are null , save their respective components on a
domain IF with suitable integer g. These sets form subsets respectively of maximal
sets of linking , critical , and invariant k-cycles, corr V9Wq, on IT9.
The first statement of the lemma has already been proved, the corresponding
sets being represented by (6.22).
To prove the second statement of the lemma let X9 be a sum of the r/th com¬
ponents of a subset of the linking cycles in (6.22). Let uq be the sum of the
(k — l)-cycles on W9 linked by the respective fc-cycles in the sums X9. The
cycle uq cannot bound below c on V9, because it would then not be spannable
corr VW, Hence X9 is a linking fc-cycle corr VqW9, and the lemma is established
for linking cycles.
Let be a sum of the </th components of any subset of the given critical
fc-cycles in (6.22). The cycle cq cannot be dependent on V9 upon fc-cycles below
c , because cq would then be 12-dependent on V on fc-cycles below c, and fail to
be a critical fc- cycle corr VW, Hence cq is a critical fc-cycle corr V9W9, and
the lemma is established for critical cycles.
Let uq be a sum of the #th components of any subset of the given invariant
fc-cycles of (6.22). If uv were not an invariant fc-cycle corr VqWq, we would
have an homology of the form
(6.23) uq ~ vq (on II9 below c )
in which vq would be a spannable fc-cycle corr V9W 9 or null. If
(6.24) vq * 0 (on V below c),
we would have uq * 0 below c, contrary to the nature of uq as an invariant fc-cycle
corr VW. If on the other hand (6.24) does not hold, vq is a spannable fc-cycle
corr VW, and (6.23) is contrary to the nature of uq as an invariant fc-cycle corr
VW. Hence (6.23) cannot hold, and uq is an invariant fc-cycle corr VqWq.
The lemma is accordingly established for invariant fc- cycles.
Suppose that a is the complete set of critical points on 12 on which J = c. Let
b be an ordinary value of J such that b is greater than c, and separates c from
greater critical values of J . The following theorem can now be established. It
depends upon the two preceding lemmas.
Theorem 6.3. A maximal set of k-cycles on the domain J < b of 12, 12- inde¬
pendent on J < b, is afforded by maximal sets of critical , linking , and invariant
k-cycles , corresponding to an admissible pair of neighborhoods VW of the critical
set <r .
Let the sets (6.20) respectively represent the maximal sets of linking, critical,
and invariant fc-cycles of the theorem. We shall first prove that any fc-cycle e
[6]
CRITICAL SETS ON S2
271
on the domain J < b of 12, is I2-homologous on J < b to a linear combination of
cycles of the sets (6.20).
Let q be the index of z, and let u q be the partition of z on IT*. According to
Theorem 5.4, uq will be homologous on the domain J(ir) < b of IF to a linear
combination of linking, critical, and invariant A:-cycles corr VqWQ. But accord¬
ing to Lemma 6.1, Unking, critical, and invariant fc-cycles corr 1 7<tW(J are ^ho¬
mologous on J < b to linear combinations of the cycles in the maximal sets (6.20)
of the theorem.
It remains to prove that the cycles in (6.20) are 12-independent on J < b.
In the contrary case there would exist a sum w of /c- cycles of (6.20) such that
(6.25) w*Q (on J < b).
Let q be a multiple of the indices of the cycles (6.20) and of the chains “involved”
in (6.25) For this q let the respective cycles in (6.20) be replaced by the cycles
(6.22) , that is, by 12-homologous linking, critical, and invariant fc-cycles corr VW }
each with all components null save one on IF. Let Wo be the sum of the cycles in
(6.22) which correspond respectively to the cycles of (6.20) in the sum w. We
have
(6.26) Wo ~ 0 (on IF, J < b).
By virtue of Lemma 6.2, the components on IF of the cycles (6.22) form subsets
of the maximal sets of linking, critical, and invariant k- cycles on IF corr VqWQ.
It follows from Theorem 5.4 that an homology such as (6.26) is impossible.
Hence (6.25) is impossible, and the cycles of the sets (6.20) are ^-independent
on J <b.
The proof of the theorem is now complete.
For reasons which we have given at length in Ch. VI and Ch. VII, we now
define the fcth type number mk of a critical set a as the number of critical A*- cycles
and spannable ( k — l)-cycles in maximal sets of such cycles corresponding to
neighborhoods VW of the critical set a. That a critical set with type numbers
mkl k = 0, 1, • • • , can be considered equivalent to a set of non-degenerate closed
extremals of J-length c, containing mk) & = 0,1, • • • , closed extremals of index
ky will be seen in §11.
From Theorem 6.3 and the definition of the type numbers of a critical set a,
we obtain the following theorem.
Theorem 6.4. Between the connectivities Pk of 12 and the sums Nk of the kth
type numbers of all critical sets of extremals we have the relations
Nk (k = 0, 1, ■••)•
In particular if Pk is infinite , Nk is infinite.
This theorem parallels Theorem 10.2 of Ch. VII.
Theorems 10.3 and 10.4 of Ch. VII likewise hold here with the interpretations
of the present chapter.
272
CLOSED EXTREMALS
[ VIII ]
The proofs of these theorems with the interpretations of the present chapter,
depend upon Theorem 6.3 and are similar to the corresponding proofs in Ch. VII.
Extremals determined by sets of cycles. In the preceding theory the critical
set of extremals has come first, and has served to determine various sets of cycles
on 12. We here reverse the process and see how a set of 12-independent Avcycles
determines a minimal set of closed extremals.
We begin with several definitions.
Let a be a critical set on 12 and V W an admissible pair of neighborhoods of a
on 12. Let c be the value of J on a. Let w be the sum of a fc-cycle below c (pos¬
sibly null) and a proper linear combination of the fc-cycles of maximal sets of
critical and linking A>cycles belonging to o-, corr VW. We term w a new cycle
belonging to <r or to the set of closed extremals determined by a. If w is
12-homologous to no cycle below c or to no new cycle “belonging'’ to a, w will be
termed a reduced new cycle and a- the corresponding reduced critical set.
Let u be a A;-cycle on 12, not 12-homologous to zero. There will exist a positive
constant c such that u is 12-homologous to no cycle below c, but is 12-homologous
to a fc-cycle below c + e, where e is an arbitrarily small positive constant.
There will be a critical set of closed extremals with ./-lengths c . We term c the
minimum critical value belonging to u. We understand that a cycle 12-homolo¬
gous to zero has no minimum critical value.
Corresponding to u there will be one or more reduced new ^-cycles 12-homolo¬
gous to ?/, belonging to reduced critical sets with the minimum critical value c.
The ensemble au of the reduced critical sets with critical value c corresponding
to all reduced new A;-cycles 12-homologous to u will be termed the minimal
critical set determined by u. Let (u) now be a set of 12-independent A;-cycles.
The ensemble of the minimal sets vu determined by all proper linear combinations
u of cycles of (u) will be termed the minimal set H of critical points determined
by (u).
The set K of closed extremals determined by points (7 r) on II will be termed the
minimal set of closed extremals determined by (w) .
Two sets of cycles ( u ) and (v) on 12 will be termed 12- equivalent if every cycle
12-dependent on cycles of ( u ) is 12-dependent on cycles of ( v ) and conversely. It
is clear from the preceding definitions that 12-equivalent sets of A;-cycles determine
the same minimal sets of closed extremals.
We continue with the following theorem.
Theorem 6.5. The sum Mk of the kth type numbers of the critical sets in the
minimal set of closed extremals determined by a finite set (u) of 12- independent k
cycles is at least the number pk of cycles in the set ( u ).
Let H be the minimal set of critical points (t) on 12 determined by (u). Let
(6.27) ci < c2 < * * * < cp
be the critical values assumed by J on H. Let <7* be the subset of H on which
[7]
THE EXTENSION OF A CHAIN ON IP
273
J — d and let ( a){ be a maximal set of new A>cycles belonging to <r». The
ensemble of cycles in the sets
(6.28) (a)i, • • • , (a)p
will be at most Mk in number. But by virtue of the definition of H each cycle of
(u) is O-dependent on the cycles of (0.28), at least if the set (6.28) be suitably
chosen. Hence
Mk ^ pk}
and the theorem is proved.
The extension of a chain on IF
7. The proof that the connectivities of the domain J < b on H are finite
depends upon certain novel consequences of our “special” homologies. We
shall now develop this aspect of the theory. We begin with a number of defi¬
nitions.
Deformation chains . The loci introduced by deformations of chains may be
divided into simplicial cells in many ways. It is essential for our purposes
that this division be made in a particular way which we shall now describe.
Let ak be an auxiliary fc-simplex, and h the line segment 0 ^ t ^ 1. We
represent the product ak X t\ by a right prism f in an auxiliary euclidean space.
We suppose ak is the base of the prism, and that a point Q = (p, t) on f is deter¬
mined by giving the point p on ak into which Q projects, and the distance t of Q
from ak . To subdivide f into simplices, we first divide it into two prisms f '
and f" by the locus / = 1/2. We then divide the prisms, f' and f", into sim¬
plices, first dividing their lateral prismatic faces in the order of dimensionality as
follows. Let f : * be a prism whose lateral faces have already been divided into
simplices. Let Z be the center of gravity of f *. We divide f * into the simplices
which are determined by Z and the simplices on the boundary of f*. In this
way we arrive at a canonical subdivision of f .
Let ak be a fc-cell on a basic complex C, and F a continuous deformation of
dk on C. We can suppose that the deformation F is defined by giving a con-
tinous point function F(pf t) of the point (p, t) on f. We understand thereby
that F(p, t) is a point on C, that F(p, 0) defines dky and that the point F(p, 0)
on dk is replaced under the deformation F at the time t by the point F(p, t).
The point function F(p, t) defines a map on C of each of the closed fc-simplices of
f . The sum of the resulting closed fc-cells onCw ill be termed the deformation
chain ak+i derived from ak under the deformation F.
The image under F of the prism f , unreduced on C mod 2, will be termed the
unreduced deformation chain derived from ak. In the sequel we shall apply
various deformations D to deformation chains H . Inasmuch as these deforma¬
tions D depend upon the unreduced deformation chains H for their definition,
it is hereby understood that the operation of reduction mod 2 is deferred until
after the deformations D are made.
274
CLOSED EXTREMALS
[ VIII ]
The deformation chain ak+\ deri ved from ak , as we have defined it, possesses the
following basic property of symmetry. If the prism f be reflected in the hyper¬
plane t — 1/2, and then mapped on C by means of the point function F(p, /),
the sum of the resulting images of the closed fc-simplices of f, viewed as a fc-chain
on C, will be “identical” with i.
The deformation <p on Kp . We understand that a point (ir) on Kp is given with
a definite ordering of its vertices
(7.0) P\ • • • , Pp.
The point (71-) on Kp and a point (x) on Jlp will be said to be K-images of one
another, if the vertices (7.0) taken in their circular order agree with the vertices
of (tt) on IT, taken in one of their two circular orders. We shall restrict our¬
selves to points (x) on Kp which define admissible elementary extremals. Let
(tt ) be such a point. As the time increases from 0 to 1 let each vertex of (w)
move along the elementary extremal which follows it on g( tt) at J-rate equal to
the J-length of that elementary extremal. Denote the resulting deformation
of Kp by <p.
The extension of a chain on Kp. We understand that a variable point
Q\ ■ • • , Qp
on Kp has the point (7.0) as a limit point on Kp , only if for each integer i on the
range 1, ■ • * , p, Q' tends to Pl as a limit point. Cells and chains on Kp are
defined on Kp with this notion of continuity. By the extension of a Ar-cell a on
K? we now mean the deformation chain on Kp derived from a under the deforma¬
tion <p. We denote this chain by Ea. By the extension Ez of a A;-chain z on
Kp we mean the sum, mod 2, of the deformation chains derived from the fc-cells
of z . If we indicate the boundary of a chain by prefixing the letter B , we see that
(7.1) BEa ss a -j- 7\a + EBa (on Kp ),
where Tia denotes the image of a on Kv under the transformation Tx. For a
chain z on Kp we then have
(7.2) BEz 3 2 + Txz + EBz (onP).
The extension of a k-chain of cells of Up. Let e be any point or k- cell of IP and
e' a point or A>cell of Kv which is a “if-image” of e. The cell e' will be unique if
and only if it is pointwise invariant under each transformation of Gp. By the
extension £e of e on IP we mean the FT-image on IP of Ee'. One sees that £e is
independent of the K- image e' on Kp used to define Ee'. This fact depends in
part upon the symmetry of the deformation chain as we have defined it.
Let w be a fc-chain of cells of IP. By the extension Zw of w we mean the sum,
mod 2, of the extensions of the fc-cells of w on IP. From (7.1) we see that
(7.3) Blw = ZBw (on IP)
since a and in (7.1) have identical K- images on IP.
[7]
THE EXTENSION OF A CHAIN ON IP
275
In proving the theorem of this section we shall make use of a deformation!?.
In defining rj it will be convenient to denote an elementary extremal with end
points Q'f Q" by (Q' Q").
The deformation 77. Let there be given a sequence of r + 1 points
(7.4) Q1, ‘ , Q'+l (r > 1)
on R such that the sum of the 7-lengths of the elementary extremals
(7.5) (QlQ2)y • • • , (QrQr+l)
is at most the constant p of §2, Ch. VTI. The deformation rj shall be so defined
as to hold the points Ql and Qr+1 fast, and deform the points
(7.6) Q\ - • • , Qr
into that sequence of r — 1 points on the extremal
(7.7) (QlQr+l)
which divide this extremal into r segments of equal 7 -length. The deformation
7/ shall also be such that the points (7.6) will be deformed in the same manner if
the points (7.4) are relettered in the inverse order.
The deformation 7? can be defined as follows.
Let h denote the broken extremal formed by the sequence of elementary
extremals in (7.5). We begin by defining a deformation rj0 of the curve h into
the extremal (7.7). Let P be the point which divides h equally with respect to
7-length . As the time t increases from 0 to 1 let two points P 1 and P2 move
away from Q on h towards Q 1 and Qr+l respectively, at ./-rates equal to half the
7-length of h. At the time t let the point on the segment of h between Px and
P2 which divides that segment in a given 7-ratio, be replaced by that point on
(P1P2) which divides (P1P2) in the same 7-ratio. Under rj0, h will be deformed
into the extremal (7.7).
We now use the deformation t?0 to define the deformation 7?. To that end let
ht be the curve which replaces h at the time fin the deformation % 0 ^ ^ 1.
To define 77 we replace the sequence of points (7.6) at the time t by the sequence
of points on ht which divide ht in the same 7 -ratios as the points (7.6) divide h at
the time t = 0. We thereby deform the points (7.6) into a sequence of points on
the extremal (7.7). This last sequence of points is finally deformed along the
extremal (7.7) into a sequence of points which divide the extremal (7.7) into
segments of equal 7-length, each point moving at a 7-rate equal to the 7-length
to be traversed. The deformation 77 is thereby defined.
We now come to an important consequence of the introduction of our special
homologies.
Theorem 7.1. If the extension Zz of a chain of n-cells of IF is an (n -f- 1)-
cycle , it satisfies the Qr-homology
(7.8) Zz * 0.
Ifz is helow c , the homology (7.8) holds below c.
276
CLOSED EXTREMALS
[ VIII ]
We shall establish this theorem with the aid of a deformation \p on K2p, and
the 2£-image ^ i of ^ on n2p. The r-fold partition of a chain w on Kp or np,
respectively, will be denoted by
prw.
The deformation \p on K2p . Let (w) be an admissible point on Kp . The
extension Et of (t) on Kp may be regarded as a curve on Kp. Consider the curve
P2Ett (on K2p)
as the continuous image of Et. The deformation \p is a deformation of p2Et on
K2p. The time in the deformation \p will be denoted by r (not t).
The curve Et is the trajectory traced by ( t ) under the deformation <p of this
section. Let (tt,) be the point thereby replacing (t) at the time t, 0 ^ t g 1.
The odd vertices of the partition p2Tt coincide with the successive vertices of
(Tt), and thus lie on g( r). The even vertices of p2nt do not in general lie on
g( x). The object of the deformation \f/ is to deform p2rt on K2p, so as to hold its
odd vertices fast, and deform its even vertices onto g(ir), carrying p2Tt into a
final image ( T°t ) which we will now descril>e.
The final image ( t\ ) of p2Tt. J^et h ' and h" be twro successive elementary
extremals of g( t). Let h i' and h" be bisected in /-lengths by points Pf and P" on
R. When t — 0 one of the even vertices Pt of p2wt will coincide with P', As l
increases from 0 to 1, this vertex P t will move from Pf to P", but not in general
on g( t). Let P* be the common end point of hr and h” . The final image Qt of
Pt under \p will nowr be defined as follows. For t fixed on the interval
(7.9) 0 g t,
Qt shall be the point on the elementary extremal (P'P*) which divides ( PfP *)
in the same /-ratio as the ratio in which t divides the interval (7.9). For t
fixed on the interval
(7.10) 1,
Qt shall be the point on the elementary extremal ( P*P ") which divides (P*P")
in the same /-ratio as the ratio in which t divides the interval (7.10). The
final image ( T°t ) of p27r* is thereby defined.
We can deform Pt into the corresponding final vertex Qh and thus deform
p2Tt into (r°t). To that end let M' and M" be the odd vertices common to
p2wt and (71-^) between which Pt and Qt lie on p2wt and ( t\ ) respectively. The
vertex Pt lies on the elementary extremal (M'M"). We use the inverse of the
deformation 77 to deform Pt into Qt> holding Mf and M " fast. The point p2 Tt is
thereby deformed into the point ( T°t)y and the curve p2Et into a curve E° tt,
all of whose vertices lie on g(r).
The definition of the deformation ^ is now complete. We observe that it
deforms points on K2p so as not to increase the value of / on the corresponding
broken extremal.
[8]
THE r-FOLD JOIN OF A CYCLE
277
The preceding deformation \ p on K2p has a deformation ^ as its X-image on
n2p
The deformation \p i on II2p. Suppose the preceding point (w) on Kv is the
X-image of a point (71-1) on IP. The curve Ew on Kp will have Crx as its /t -image
on np. The iC-image of the curve p2Eir on K2p will be the curve p2Ewx on Il2p.
The deformation \p will have as its K-image on JI2p a deformation ^1 of p2Et}
into the K-image on IT2p of E° t. Denote this i£-image of E°tt on IP7> by E°t.
The deformation thereby defined will be independent of the particular
Tv- image of (tj) used to define ^ on K2p.
The curve p2E tti and the curves replacing p2Ettx under \px are closed curves on
II2p. Moreover the final curve E°7rx, regarded as a 1 -chain on II27', reduces to
zero , mod 2. This follows from the fact, that on K2p the application of Tx to
the 1-cell traced by (71-^) as t increases from 0 to i, yields the 1-cell traced by
(7 r°t) as t increases from \ to 1 .
With the deformation defined as above, the main body of the proof of
Theorem 7.1 can be incorporated in the following lemma.
Lemma 7.1. The 2-fold partition p2Ez of the cycle Ez of the theorem , can be
J -deformed on n2p into a set of n-cells whose sum reduces to zero , mod 2.
To prove this lemma we regard each point of z as typified by the point {n\)
on IP' used in the definition of \px. We then regard p2Ez “unreduced” as a locus
of the curves p2Eirh and the cells of p2Ez as the images of the cells of the deforma¬
tion chain Ez, each point on Ez corresponding to its 2-fold partition on p2Ez.
But under \px each curve p2E tti is deformed on IP7' into a set of 1 -cells which sum,
mod 2, to zero. By virtue of our canonical division of a deformation chain it
follows that the set of ( n + l)-cells into which p2Ez is thereby deformed on
H2p likewise sum, mod 2, to zero. The proof of t he lemma is now complete.
To turn to the theorem, recall that
(7.11) Ez * p2Ez
by virtue of our special homologies. But according to the preceding lemma,
(7.12) p2Ez ~ 0 (on II2p).
From (7.11) and (7.12) we see that (7.8) holds as stated.
The r-fold join of a cycle
8. The r-fold partition of a point (7 r) on IP always exists. It is a point on
nrp. On the other hand an arbitrary point on IIrp is not in general the r-fold
partition of any point on IP. The process of taking the partition of a point on
IP does not then admit an inverse applicable to all points on nrp. Nevertheless
there is another process applicable to a limited class of chains on nrp which for our
purposes takes the place of an inverse of a partition. The chain on IP which
is thereby made to correspond to the given chain zrp on nrp is termed the join
278
CLOSED EXTREMALS
l VIII ]
of 2 rp on IF. In this section we shall define and analyse the join of a chain on
llrp. The results obtained are fundamental in our final theory of 12-homologies.
The complex 0. Let r be an integer greater than 1. Let Gq denote the sub¬
group of G9 generated by the transformations T r and f/0. The complex IF was
formed from Kq by identifying the cells of A '' which were the images of one
another under transformations of the group Gq. Let 0 be the complex similarly
formed from Kq by identifying the cells of Kq under the transformations of the
group Gq.
A point (7 r) on 0 represents a class of points (x') on K<1 obtainable from any
member of the class by means of the transformat ions of Gq. Points of the class
(x') on Kl and the corresponding point ( x ) on 0 will be termed A"-images of
one another. Points (x) and (x") on 0 and IP respectively which possess a
common A-image on Kq will likewise be termed A-images of one another.
The r-fold join on IF of a A-eyde zrp on Ilrp will be defined only for those
k- cycles which satisfy the following two conditions.
A. The k-cycle zrp on IIrp shall he the K-image, reduced mod 2, of a k-cycle
wrp on 0.
B. The points (x) on zrp shall determine curves g(n) on which the J -lengths of r
successive elementary extremals is at most p.
To define the r-fold join of zrp let (x) lx* a point on wtp, and
(8.1) P\P\ - , Pq (q = rp)
the vertices of a point on Krp which is the A-image of (x). By arbitrarily
preferring the vertices
(8.2) Pr, P~r, ,P’P
we obtain a point on Kp. Let <p(x) denote the A-image on TF of the point
(8.2) on Kp. The point <p(x) is uniquely determined by (x). That is, it does
not depend upon the particular A-image (8.1) which is selected to represent (x)
on Krp. For any other A-image on Krp of (x) would be obtainable from the
point (8.1) by applying a transformation of the form Tmr or TmrU 0 to the point
(8.1), and would accordingly lead to the same point <p(x) on IF. It follows that
the points <p(x) form a continuous image on IF of wrp. Reduced, mod 2, this
image is a fr-cvcle on IIP which we term an r-fold join of zrp determined by wTp.
We shall now prove the following theorem.
Theorem 8.1. A k-cycle zrp which possesses a join zv on IF satisfies the
$l-homology
(8.3) 2F * 2".
This 9,-homology can in particular he realized by using the deformation 77 to deform
zrp into the r-fold partition przp of zp on Urp.
Let (x) be a point on wrp and (x') the corresponding point ^(tt) on a join zp of
zrp . Let Pf and P" be two successive vertices of (x'). Holding Pf and P" fast
[8]
THE r-FOLD JOIN OF A CYCLE
279
we can use the deformation rj of §7 to deform the vertices of (71-) between P' and
P" into the correspondingly ordered vertices of pr7r' between P' and P". We
thus have
and
zrp /™s^ J)TZP
(on IIrp)
zp * pTzp
from which (8.3) follows as stated.
We shall say that an n-cycle z q is simple if it possesses the following properties.
Each (n — l)-cell is incident with just two ?*-cells. The n and (n — l)-cells
incident with an (n — 2)-cell form a circular sequence in which n and (n — 1)-
cells alternate, and in which each n-cell is incident with the preceding and
following (?i — l)-cells and no other (n — l)-cells of the sequence.
We have given conventions under which the cells of a sum shall be regarded
as identical. If these conventions are made optional so that cells previously
regarded as identical may or may not be regarded as identical at pleasure, the
resulting chain will be termed an unreduced chain. With this understood it is
clear that any ??,-cycle can be replaced by an unreduced n-cycle which consists'
of the same n-cells, but for which the conventions of identity relating to the
cells of lower dimensionality have been so altered that the new cycle is simple.
We now state an important lemma.
Lemma 8.1. Let zq be a “ simple ” k -cycle of cells of IF, q — rp , no cell of which
has a K-imagc on K<] invariant under a transformation of Gq other than a power of
T 2r- There then exists an unreduced (fc — 1 ) -chain y q of cells of zq such that the
chain
(8.4) zq + lyq {on IT7),
unreduced mod 2, is the K- image of a cycle wq on 0.
We shall first define the fc-chain wq on 0, and after deriving certain properties
of wQ obtain a chain yq with the required properties.
Definition of wq. Corresponding to each closed fc- cell ak of zq let an arbitrary
iwmage hk be chosen among the closed /c-cells of Kq. We form the sum
(8.5) m* = 2bk (on Kq)
of these cells. Let us denote the J^-image on 0 of a chain z on Kq by 02. The
chain w q will be defined as a sum
(8.6) wq — ®{uk -f Evk~\) (on 0)
where vk~\ is a (fc — l)-chain on Kq still to be defined, and Evk~ 1 is the extension
of t;jfc«ion K9.
The chain vk~i shall consist of a sum of (fc — 1 )-chains avk-x on K9*, one corre¬
sponding to each (fc — 1 )-cell ak~\ of zq.
280
CLOSED EXTREMALS
[ VIII ]
Definition of avk^x. On the closed A>cells bk in the sum (8.5) let b'k and b"k
denote the K-i mages of the two £-cells incident with a*- 1 on zq. Let bk^x and
bjfc-! be the if-images of ak-i on the boundaries of bk and bk respectively. There
will then exist an integer m between 0 and q — 1 inclusive, such that one of the
two following relations holds.
(8.7)
C, = Tmb'k-i,
Case I;
(8.8)
bl- 1 = U«TX-u
Case 11.
We then define
by the congruence
(8.9) al _
-1 — T0 S Ar— i + Tjbfc-x + • • • + Tm_]6A.„]
(on K")
understanding that the right member is null if m = 0.
For the sake of brevity we write (8.9) symbolically in the form
al- i = (T0 + ■ • * + .
We introduce the symbol
T\ = T{ + T{+t + • • - + Tu
understanding that T\ is null iij < i. We then write (8.9) in the form
(8.10) al S TT%-i-
The chain BEvk„x in (8.6) shall consist of a sum of A;-chains, each of the form
(8.11) BET'S-1 6*-i (on 0),
one corresponding to each ( k — l)-cell a*_i oizq.
We shall now prove the following statement.
(a). If one interchanges the rdles of bk-x and bk-x in the definition of aJL1? the
chain (8.11) is unaltered, mod 2.
We first suppose that Case I holds. Let us interchange the roles of bk-x and
and put (8.7) in the form
bit-! -
Here £ = 0 if m — 0. If m = 0, (a) is clearly true. If m > 0, — q — m and
we proceed as follows.
The new chain (8.11) will be the chain
QETtr*-xbl-l9
which may be written in the form
(8.12) eET'^K-i
upon usingc(8.7). The chains (8.12) and (8.11) accordingly have the sum
(8.13) eETr'K-i.
[8]
THE r-FOLD JOIN OF A CYCLE
281
Now two chains such as
(8.14) QETM-u @ETi+rbl~i
are equal, mod 2, on ©. But q is an even multiple of r by hypothesis, and the
chain (8.13) accordingly involves q/2 pairs of chains such as the pair (8.14).
Thus (8.13) reduces to zero, mod 2, and statement (a) is proved in Case I.
In Case II, we first rewrite (8.8) in the form
(8.15) Ci = U0Tmb'k-i
as is possible since U 0 Tm is its own inverse. Having thus interchanged the roles
of bk„1 and &*_!, the chain (8.11) is replaced by the chain
(8.16) ®ETTl Ci-
Upon using (8.8), chain (8.16) takes the form
(8.17) BETT'U'TX-i-
To reduce (8.17) to the form (8.11), observe that
T{Uo = UoT-i
so that (8.17) becomes
(8.18) ©tfl/077Ci.
Reference to the definition of E shows that if w is any chain on Kq ,
(8.19) EU0w = UoET-xw,
so that (8.18) takes the form
(8.20) QUoErr%-i.
Finally for any chain w on Kq
QU0w — ©w,
so that (8.20) reduces to the chain
(8.21) ©tfrj^Ci,
and is thus equal to the chain (8.11) as stated.
The proof of (a) is now complete.
To replace a cell w on K 9 by U 0w will be termed changing the sense of w . We
shall now prove the following.
(b). The senses of the cells bk and bk can be separately or jointly changed at
pleasure , without changing the chain (8.11) corresponding to a*_ x.
We shall first establish (b) for the case in which the cells b'k1 b”k are replaced by
the cells
282
CLOSED EXTREMALS
[ VIII ]
The cells b'k^l and bl will then be replaced by the cells
(8.22) Ci = Cn Ci = UJbl- 1.
Suppose now that Case I holds for b [„1 and bl . From (8.7) and (8.22) we
see that
Pk- 1 — U<sTmfik-\)
so that Case II holds for /Ci and fil-i • One sees that the chain (8.11) remains
unchanged.
Suppose now that Case II holds for b 'k _ t and blc^1 . From (8.8) and (8.22) we
find that
Pk -1 ~ ^mPk-l
and we see that the chain (8.11) is again unchanged.
Finally I say that all other changes of sense of bk and bk reduce to the cases
just considered. For by virtue of (a) the r61es of bk and bk can be interchanged
without changing (8.11). Moreover to change the senses of bk and bk jointly
it is sufficient to change their senses separately in succession, thereby producing
no change, mod 2, in (8.11).
Statement (b) is thus established.
Let z be a chain on a given domain. The boundary of z on the same domain
will be denoted by Bz. With this understood we shall prove the following
statement.
(c). The boundary of the chain wq on 0 is the cycle
(8.23) Bwq s Z'SETr'Bbl-i
where the sum 2* contains one term corresponding to each ( k — \)-cell ak~ i of zq and
where bk^v and m are determined with the aid of ak- 1 as previously .
From (8J3), (8.6), and (8.11) we see that
(8.24) 206* -f 2*0i?77“'16*-1,
where the terms in the sums 2 and X* correspond respectively to the k- and
(k — l)-cells of zq) and are summed for all these cells. We note the relation
(8.25) B&u s &Bu (on 0)
where u is any A>chain on Kq. From (8.24) we then find that
(8.26) Bwq s X@Bbk + X'QBETr1 Ci •
To evaluate the sum 2*, we refer to (7.2) and see that
bett'K-i = tt’&Lx + rrfiLi + ebtt'K-i
=* + TX-, + ETT'BhX.
[8]
THE r-FOLD JOIN OF A CYCLE
283
Upon using (8.7) or (8.8), according to the case in hand, we find that
(8.27) <dBETrlbi x - ©Ci + ©Ci + QET™”1 Bb'k~1.
Independently of the preceding, we note that
(8.28) xmbk s 2*i@Ci + ©CJ,
summing as in (8.24). With the aid of (8.26), (8.27), and (8.28) we obtain (8.23)
as written.
Statement (c) is thereby proved.
We continue with a proof of the following.
(d). Let ak-2 be an arbitrary (k — 2)-cell of zq. The subset of (k — \)-cells on
the boundary of wq obtained from (8.23) by omitting atl ( k — 2)-cells of Bbk^.l save
those which have ak~ 2 cls a K-image on II9 sum to zero , mod 2. Hence Bwq s= 0
on 0 and w q is a cycle.
Recall that z q is a simple A>cycle. The A> and ( k — l)-cells of z q incident
with a,k~ 2 taken in their circular order about ak~2 will be denoted by
(8.29) a[l) a[-\ a[i} all\ • • • a(k" ' a^\ (on II*).
In forming the sum (8.5) we have selected A-images on Kq of the respective
/c-cells of zq. Using these same A-images on Kq, let the respective A-i mages of
the /c-cells in (8.29) be denoted by
(8.30) b[l )b{r> ■ ■ • &<**> (on if*).
Let
blLY, biLY (* = 1,2, ••• ,s)
be the i£-images of akl\ on the boundaries of b\' 1 and b'k'" 1 ' respectively, under¬
standing that h*s+1) = 6*1 ■ By virtue of (c) we will lose no generality if we
suppose the senses of b,2 • • • , b)‘ 1 have been successively changed so that
(8.31) bY2\ = TmmbHY (* = 1, • • ■ , « - 1)
where m(t) is an integer between 0 and q — 1 inclusive.
Case II thus does not then occur corresponding to ak!i\, for i = 1, • * ■ , s — 1.
Nor will Case II then occur corresponding to akl\ as we shall now prove.
To that end let fi(l) be the A-image of a*_2 on b(k\ i = 1, • • , 5. Observe that
/3(,+1) and /3(t) lie respectively on the boundaries of bk and bkl\' . From (8.31)
it then follows that
(8.32) 0(i+1) = Tm(o/3(i) (t = 1, * * • , * - 1).
If akL\ came under Case II, we would have a relation of the form
(8.33) p™ - U0TmwV> (0 g S g - 1),
and we could infer from (8.32) and (8.33) that
(8.34) 0(1) = UoTrf™
284
CLOSED EXTREMALS
[ VIII ]
where
(8.35) y = m(1) + * ' * + m(t)-
But (8.34) is contrary to an hypothesis of the lemma. We infer that a£i\ comes
under Case I as stated.
The relations (8.32) may now be completed by the relation
(8.36) 0(1) = Tm^K
Relations (8.32) and (8.35) combine into a relation
(8.37)
where y is given by (8.35). By virtue of the principal hypothesis of the lemma
we see that y has the form
(8.38) y = 2^r
where v is an integer, positive, or zero.
To return to statement (d) we observe that the terms in the sum 2* of (8.23)
corresponding to the ( k — l)-cells of z 9 incident with a*- 2, take the form
«
(8.39) 2 QETf'-'BblLY (on 0).
t = 1
To establish statement (d) we omit all of the ( k — 2)-cells of Bb l L\' save
With this omission (8.39) reduces to the chain
9
(8.40) 2
t = 1
Upon using (8.32) the chain (8.40) takes the form
(8.41) @ETrlP(1) * SE(T0 + Tx + • • • +
where y is given by (8.35). But, on 0, pairs of chains of the form
QETJM, eETm+r0W
are equal, mod 2. Since y — 2 vr, there are v such pairs in the sum (8.41). The
chain (8.41) accordingly reduces to zero, mod 2.
Hence the boundary of wq on © reduces to zero, mod 2, and (d) is established.
We now return to the proof of the lemma. On K 9 we set
Uk = 2)5*,
Vk-i = 2*7T%- 1.
We see from (8.24) that the cycle wq on © takes the form
u>9 = @(uk + Evk _,).
[9]
FINITENESS OF THE BASIC MAXIMAL SETS
285
Now zq is the if -image on n9 of uk. We let yq be the if-image on n9 of vk-i
unreduced mod 2. We observe that yq is a sum of ( k — l)-cells of zq. The
cycle w q on © is then the i^-image on n9 of the cycle
2 9 + Zyq (on n9),
unreduced mod 2.
The lemma is thereby proved.
We can now prove the following theorem.
Theorem 8.2. Let zq ( q — rp) be a k-cycle of cells ofl I9 ( below c), no cell of which
has a K-image on Kq invariant under a transformation of Gq other than a power of
Tir. Suppose also that points (tt) on zq determine curves g(ir) on which the J-
lengths of r successive elementary extremals are at most p. Then zq is 12- homologous
(below c) to a k-cycle on II p.
By virtue of the preceding lemma there exists a (k — l)-chain yq of (k — 1)-
cells oi zq such that
(8.42) zq + Zyq (on II9)
unreduced mod 2, is the if -image of a A;-cycle on 0. The fc-cycle (8.42) accord¬
ingly possesses a join xv on Hp. According to Theorem 8.1 we have
(8.43) zq + Zyq * x *
below c, if s q is below c. But Zy q is a cycle on II9 since z Q + Zy Q is a cycle, and,
by virtue of Theorem 7.1,
(8.44) Zyq * 0.
Hence
(8.45) zq * xv
below c if y q is below c. The proof is accordingly complete.
Note. For the sake of reference it is important that we have more intimate
knowledge of how the homologies (8.43) and (8.44) are generated. Reference
to the proof of Theorem 7.1 shows that (8.44) is the result of applying the
deformation to the 2-fold partition of Zyq. Reference to Theorem 8.1 shows
that (8.43) is effected by using the deformation 77 to deform the cycle (8.42) into
the r-fold partition of xv on n9.
Finiteness of the basic maximal sets
9. Let <r be a critical set on 0 on which J = c. Some of the closed extremals
determined by a may be multiply covered. In case a closed extremal 7 possesses
at most a finite number of multiple points, and is covered v times by a closed
extremal gy we shall say that g possesses the multiplicity v. A given critical
set may possess closed extremals with several different multiplicities.
We shall now prove a lemma which has immediate bearing on the finiteness of
286
CLOSED EXTREMALS
[ VIII ]
maximal sets of spannable and critical A*-cycles corresponding to an admissible
pair of neighborhoods VW of a.
Lemma 9.1. Corresponding to the critical set <r there exists a positive integer p
with the following property. If VW is an admissible pair of neighborhoods of cr,
any k-cycle on W ( below c) is Q-homv logons on V ( below c) to a, J -normal k-cycle
which is arbitrarily near a, and whose components are null except at most its com¬
ponents on Wp.
We shall prove that the lemma is satisfied by any integer p such that pp > c,
and such that p is an even multiple of the multiplicities of the closed extremals of
the given critical set.
Let p be such an integer. Let u be any k-cycie on W. Let q be a multiple
of p and the number of vertices in the non-null components of u} say q — rp.
Without loss of generality we can suppose that the components of u are so near a
that u possesses a partition zq on IP7. Without loss of generality we can also
suppose that zq is composed of ./-normal points on IP', because in any case such a
cycle would be obtained from zq by an application of the deformation 0q(i).
We shall now investigate the applicability of Theorem 8.2 to zq. We shall
first verify the fact that ./-normal points on WQ sufficiently near a possess no
A-imagos on W 9 invariant under transformations of Gq , other than powers of
T<2r, where q — rp.
Let (7 r) be any ./-normal point of oq. Suppose that g( r) has the multiplicity
v. There will then be q — sv vertices in (7 r), where s is a positive integer. It is
clear that a AMmage of (t) on A' will be invariant under no transformations
of Gq other than powers of T *.
From the fact that
q — rp — sv
and that p is an even multiple of v we infer that $ is an even multiple of r. Thus
A"-images of (n) on Kp are invariant at most under powers of T2r. Finally it is
clear that /-normal points sufficiently near a will have this same property.
Without loss of generality we can then assume that zq is so near o that the
A-images of its points are invariant at most under powers of T2r.
In order to apply Theorem 8.2 to zq we must know that its points (w) define
curves g( w), the /-lengths of r of whose consecutive elementary extremals is at
most p. If (7r) is a /-normal critical point of <jq, the /-length of r of its elemen¬
tary extremals is
r
c
Q
P-
Moreover if z q is a /-normal cycle sufficiently near <r, the /-length of r of its
elementary extremals will still be less than p.
Theorem 8.2 is thus applicable to zQ provided z q is a fc-cycle of cells of II9.
But if zq is not a k-cycle of cells of IT, upon subdividing IT9 and zq sufficiently
[9]
FINITENESS OF THE BASIC MAXIMAL SETS
287
finely, one can obtain a cycle uQ of cells of IF homologous to zq below c if z is
below c, and so near zq that Theorem 8.2 is applicable to uq . We conclude
that zq is ^-homologous to a &-cycle on IF. Preference to the note following
Theorem 8.2 makes it further clear that if 2 q is sufficiently near aq, as we suppose
it is, uq and hence z q is ^-homologous on V to a fc-cycle on Wp, the homology
holding below c, if z q is below c.
Finally any fc-cycle on Wv is homologous on Vp (below c) to a /-normal fc-cycle
on Wp arbitrarily near av. The proof of the lemma is now complete.
We now establish an important consequence of the preceding lemma.
Theorem 9.1. There exists at most a finite number of spannable or critical k -
cycles corr VW in maximal sets of such cycles.
We shall give the proof of the theorem for the case of spannable k- cycles.
Let z be any spannable k- cycle corr VW. According to the preceding lemma
z is 12-homologous on V below c to a k- cycle on Wp where p is a positive integer
dependent only on a. But there are at most a finite number of A;-cycles on Wv
below c, independent on Vp below c. The theorem is accordingly true for the
case of spannable k- cycles.
The proof for the case of critical ^-cycles is similar.
The following theorem is an easy consequence of the final statement in
Theorem 4.1.
Theorem 9.2. If b is a positive number less than the least critical value of J, the
SI- connectivities of the domain J < b are null.
For if z is a fc-cycle on J < 6, any non-null component of z on IP is homologous
to zero on IF, according to Theorem 4.1, bo that z is ^-homologous to zero on
J < b as stated.
We conclude this section with the following theorem.
Theorem 9.3. If b is any ordinary value of J, the 11- connectivities of the domain
J < b are finite .
We have already seen that the number of A;-cycles in maximal sets of spannable
or critical A;-cycles corresponding to a critical set <7 is finite. The number of
linking A;-cycles in a maximal set corresponding to any complete critical set cr is
then finite, for it is at most the number of spannable ( k — l)-cycles in a maximal
set corresponding to a.
To establish the theorem we let
C\ C2 "C * ^ Cm
be the critical values of J less than b , and let
&1) °”2j ' > 0m
be the corresponding complete critical sets. There are no invariant fc-cycles
corresponding to cx and <rh since there are no fc-cycles below cx except those
^-homologous to zero. Let (3X be a constant such that cx < ($1 < c2. According
288
CLOSED EXTREMALS
[ VIII ]
to Theorem 6.3 a maximal set of A;-cycles on the domain J < ph 12-independent
on this domain, will be afforded by maximal sets of critical, linking, and invariant
A;-cycles corresponding to the critical set Since these maximal sets are finite,
the ^-connectivities of J < Pi are finite.
We now assume the theorem is true for any domain J < pr for which cr < pr <
cr f i, and prove it is true for the domain J < pr+i, reasoning as in the preceding
paragraphs. It follows by mathematical induction that the theorem is true as
stated .
Numerical invariants of a closed extremal g
10. In this section we shall define the index and nullity of a closed extremal g
in a way that will be independent of the coordinate systems used to cover the
neighborhood of g. We first introduce two important conceptions.
Proper sections S of Rp belonging to g. Let c be the /-length of g} and p a
positive integer such that pp > c. Let (7 r0) be an inner point of IIP such that
g(ir0) = g. Suppose that none of the elementary extremals of g(i r0) reduce to
points. Let
Pi (9 = I,*", V)
l;e the qth vertex of (7r0), taking these vertices in one of their two circular orders.
Let Mq be a regular analytic (m ~ l)-manifold intersecting g at PJ, but not
tangent to g. A manifold of points (t) whose gth vertex Pq is subject to no
other restriction than to lie on Mq neighboring P J will be called a proper section
S of Rp belonging to g.
The boundary problem associated with S. With g and S we shall now associate a
boundary problem of the type studied in Ch. V. To define such a problem we
cut g at P J , forming thereby an extremal segment 7 of J- length c, with end points
A 1 and A 2 wrhich are copies of P£ . Let (x) be an arbitrary coordinate system on
R neighboring PJ. We shall regard the points A 1 and A2 on 7 as distinct, and
shall provide them with neighborhoods which we shall also regard as distinct.
These neighborhoods will be represented by copies of the coordinate system
denoted by xn and x'2 respectively. We suppose that Ml is regularly represented
in the form
(10.1) x 1 = xi(ah •••,<*„) (n = m — 1)
and that Pj corresponds to the parameter values (a) = (0). With g and S we
now associate a boundary problem B in which the end conditions refer to points
neighboring A1 and A2 respectively, and have the form
(10.2) xis = x{{a) (s — 1, 2; i — 1, • • • , m)
where the functions z*(a) are those defining M1. We see that 7 will be a critical
extremal in the boundary problem B.
Let ( v ) be a set of parameters in a regular analytic representation of S neigh¬
boring (ttq) . Suppose that ( v ) = (0) corresponds to (x0). On S the value of
[10]
NUMERICAL INVARIANTS OF A CLOSED EXTREMAL g
289
at the point (w) determined by ( v ) will be a function f(v), analytic in (v)
at ( v ) — (0). The point ( v ) = (0) will be a critical point of/(t>). The form
(10.3) Q(v) = fvtv}iO)viVj ( i,j = 1, • • • , pn)
will be an index form corresponding to 7 as a critical extremal in the preceding
boundary problem B.
We shall now prove the following theorem.
Theorem 10.1. The index and nullity of the form (10^3) are independent of the
proper section S of Rp on which the function J — f( v) is defined.
Suppose g has the length go. We shall combine the conditions (11.2a) and
(11.2b) of Ch. V into a system
(10.4a) yT = 0,
(10.4b) Ltiri) + = 0 (i = 1, • • • , m).
We admit solutions of the system (10.4) in the form of contravariant tensors
locally of class C 2 in terms of the arc length t along g. Recall that rjT is an
invariant, and that the left member of (10.4b) is a tensor which is covariant with
respect to admissible changes of coordinates ( x ) along g. In the system (10.4)
we are free from the necessity of having a single coordinate system along gy and
in particular free from the difficulties which arise in connection with such co¬
ordinate systems when R is non-orientabie.
The theorem is a consequence of the following lemma.
Lemma 10.1. The index of the form (10.3) equals the number of solutions of the
system (10.4) of period go which are independent of tangential solutions of (10.4),
and correspond to negative values of X. The nullity of the form (10.3) equals the
number of solutions of (10.4) of period co which are independent of tangential solu¬
tions of (10.4), and correspond to a null value of X.
Corresponding to the extremal segment 7 and the end conditions (10.2), the
accessory boundary problem (11.2) of Ch. V here takes the form
(10.5a) rjT = 0 (s = 1, 2; i = 1, • • • , m),
(10.5b) Li(r,) + Xt?" = 0,
(10.5c) rf - x'kuk = 0,
(10.5d) xi(!\ - r?) = 0 (A, k = 1, - • - , n = m - 1),
where x%h is the partial derivative of x\a) with respect to a*,, evaluated for (a) =
(0). According to Theorem 14.1 of Ch. V the nullity of Q(v ) will be the index of
X = 0 as a characteristic root of the system (10.5), and the index of Q(v) the
number of characteristic roots of (10.5) which are negative, counting these roots
with their indices.
290
CLOSED EXTREMALS
[VIII]
To compare the system (10.5) with the system (10.4), first recall that the
manifold (10.1) is not tangent to g. The end conditions (10.2) are then seen to
satisfy the non-tangency condition of §12, Ch. V. It follows from Lemma 12.1
of Ch. V that the system (10.5) admits no non-null tangential solutions, so that
no proper linear combination of independent characteristic solutions of (10.5)
is a tangential solution.
We shall now show that each characteristic solution (77) of (10.5) has the period
cx) i n t.
First we note that for such a solution
(10.0) 17 1“ - vil (1 = 1, • • • , ro)
as follows from (10.5c). It remains to prove that
(10.7) r2 = ri.
To that end let x* = 7 '(t) be a representation of g neighboring P£, measuring
the arc length t from Pj. Upon using the definition of as together with
(10.6), we set' that
(10.8) 7'(0)(d - 0) = 0.
We combine (10.8) with (]0.5d) to form the system
7f(0) (f • - 0) = 0,
(10.9
* (0) (f - 0) = 0.
But the m~ square determinant
7 * (0)
**(0)
(h = 1, • • • , n; i = 1, • • • , m)
is not zero since the manifold xl — x\a) is not tangent to g when (a) = (0),
From (10.9) we conclude that (10.7) holds, and hence that each characteristic
solution of (10.5) has the period « in t.
The lemma and the theorem follow directly.
We now define the index and nullity of g as the index and nullity of the index form
(10.3) determined by any proper section S belonging to g of the space Rv . If the
nullity of g is zero , we term g non-degenerate.
We add the following theorem.
Theorem 10.2. If g is a non-degenerate closed extremal , there is no connected
family of closed extremals which contains both g and closed extremals other than g.
If the theorem were false, there would be a connected family of critical ex¬
tremals corresponding to the boundary problem (10.2) which would contain g
and closed extremals other than g. We would then have a contradiction to
Theorem 11.1 of Ch. VII. We accordingly infer the truth of Theorem 10.2.
[ 11 ] THE NON-DEGENERATE CLOSED EXTREMAL 291
The non-degenerate closed extremal
11. In this section we shall prove that the 2th type number of a non-degenerate
closed extremal g of index k is 8kiy i = 0, 1, • • • . We shall accomplish this by
showing that the type numbers of the critical set a defined by g on il are the
same as those of g considered as a critical extremal, in the sense of Oh. VIT, in a
boundary problem B with end conditions of the form (10.2).
Let (tt) be an arbitrary point on TP. Let (x') be any point on IP whose
vertices lie on g( x) and possess a circular order in agreement with their order on
<?(x). With (x) and (x') we now associate a third point (tt") on the extension
L 7r of (7 r).
The mean correspondent of (tt') on £7 r. Let g * be an unending curve covering
<7(71-). On g* let * represent the ./-length measured in a prescribed sense from a
prescribed point on g*. The vertices of (tt') will be represented by infinitely
many copies on g*. Let
(11.0) P\ .. ,P*
be a set of copies of the p vertices of (tt') which appear consecutively on g* in the
order (1 1 .0). Let
(11.1) Q\ ••• ,Q*
be a set of consecutive points on g* which define a point (x") on £ V such that the
average value of ,s for the points (11.1) is the same as for the points (1 1 .0). We
term (7 r") the mean correspondent of (7 r') on £x.
We observe that the mean correspondent of (tt') on f 7r will be independent of
the particular set of vertices (11.0) chosen to represent the point (x'), and of the
sense assigned to g(x), as well as of the point on g(ir) from which ,9 is measured.
The deformation F p. We now deform (x') into its mean correspondent (x")
on fx moving each vertex Pl along g* to the corresponding vertex Q\ moving
Pi at a ,7- rate along g* equal to the 7-distance to be traversed. We term this
the deformation Fv.
Let c be the ./-length of g and p a positive integer such that pp > c. Let <rp
be the critical set of ./(x) determined by g on IP, and a the corresponding critical
set on Q. Let S be a proper section of Rv belonging to g, as defined in §10.
Let q be a second integer such that qp > c . Let (x0) be a point of a 9 none of
whose elementary extremals are null. If (x) is a point on II9 sufficiently near
(x0) , there will be a unique point bp{r) on S with vertices on <7(x). We term
bp(x) the extremal projection of (x) on S, and state the following lemma.
Lemma 11.1. Let S be a proper section of Rp belonging to g . Corresponding to
S and an arbitrary neighborhood N of a on S2, any J -normal k-cycle zv on IP ( below c)
sufficiently near crp will possess an extremal projection bp(zp) on S such that
(11.2)
2P * bp(zp)
[on N ( below c)].
292 CLOSED EXTREMALS [ VIII ]
If zp lies on a sufficiently small neighborhood of <rp , it will possess the properties
enumerated in the following paragraph.
The extension Zwp of any cycle wr on zv will satisfy the relation
(11.3) Zwp * 0 [on N (below c)]
as follows from Theorem 7.1. The extremal projection bp(zp) of zv on S will exist,
and be the continuous image of zp, points on zp corresponding to their extremal
projections on S. The extremal projection bp( x) of any arbitrary point (x) on
zp will possess a mean correspondent /3p(x) on £x which varies continuously
with (x) on zp. As (x) ranges over zp, its image pp(ir) will define a cycle Pp(zp),
the continuous image of zp. Moreover the deformation Fp of bv( x) into /3p(t)
will deform bp(zp) continuously on Np (below c) into Pp(zp). Accordingly
Pp(zp) ~ bp(zp) [on Np (below c)].
We regard the deformation chain Zzp , unreduced mod 2, as the product of zp
and a circle. We see that pp(zp) is a singular fc-cycle on Zzp . According to the
theory of product chains, of which Zzv is an instance, there will exist a (k — 1)-
cycle wp on zv , such that
(11.4) pp(zp) ~ zp + £™p (on Zzv).
The lemma now follows from the three preceding homologies.
Note. The homology (11.4). For the sake of an application in Ch. IX we
shall here exhibit a (k + l)-chain on Zzp bounded by the members of (11.4).
According to the definition of the extension of a chain, Zzp , unreduced mod 2,
can be regarded as the product of zp and a circle whose parameter t represents the
time in the deformation defining Zzp, with 0 ^ K 1. Any point (x') on Zzv is
thereby determined by a pair (x, t) in which (x) is a point on zv and t a value of t
on the interval 0 S t < 1. It will be convenient to regard the point (x') as
represented not only by the pair (x, t), but also by all pairs of the form (x, t + n)
where n is any integer, positive, negative, or zero. We have thereby covered
Zzp by an unending succession of copies of Zzp in the form of a product W of zp
and the unlimited t axis. On W, z p is represented by chains corresponding to
integral values of t.
Let a* be any i-cell of zp. Let be the corresponding z-cell of /3p(zp). If zp is
sufficiently finely divided, as we suppose it is, the cell fe* will be represented on
W by at least one and at most two continuous image cells b) and b], whose
closures lie on W at points at which
0 < t £ 2.
If there are two such cells b\ and b2i} the values of t on one such cell, say b\, will
exceed the values of t at the corresponding points on the other b\, by unity.
Let (x) be any point on at- and (x, tl ) the corresponding point on b). With
the point (x) on a{ we now associate the closed 1-cell on W of points (x, t) for
which
0 g t S t\
THE NON-DEGENERATE CLOSED EXTREMAL
293
l 11]
The ensemble of these 1-cells on W as (?r) ranges over at will be denoted by w)±x ,
and termed the first associate of a*. If b\ exists, we similarly define a second
associate w]+x of aiy replacing t1 by tl + 1.
We shall now describe a division of the point sets w\+1 and x into cells.
First suppose that on w\ + lf
0 ^ t S 1.
If a, is any boundary cell of a* and w) + x its first associate, w)+x will lie on the
(geometric) boundary of w\¥X. We proceed inductively, supposing that w)+1
has already received a division into cells. In case j — 0, w)+l shall consist of
two consecutive 1-cells, the point of division being arbitrary. For any j the
geometric boundary of w)+x will include an image of a, on w) + x consisting of
points at which t — 0, and an image of the corresponding cell b), the locus of the
other extremities of the 1-cells making up w j. We suppose that the cells of
these images are included among the cells of w ) + x . We now choose an arbitrary
inner point P on w\+x, and join P by suitable “straight” cells to the boundary
cells of w\+ x , thus completing the division of w\.{ x into cells in case 0 ^ t ^ 1 on
^i+i-
A second associate of ax will exist if and only if 0 g t ^ 1 on w\+x . In such a
case we let w\+x denote the copy of w\+x on W obtained by adding 1 to the
parameter t in the pair (r, t) which represents an arbitrary point on w\+x . Let
£id, represent the closure of an image of la% on W on which 0 | | 1. We
suppose £] Ld» has the division into cells which we have accorded a deformation
chain. We now give w2i+x a division into cells such that
(11.5) w\+i = £i<L -j- w »-m
becomes a valid congruence.
In case there are points on a set w \ + x at which t > 1, there may be a cell a, on
the boundary of ax such that w*+ x is on the geometric boundary of w\+i. We
then divide such sets w)+x into cells in the order of their dimensionality in acT
cordance with (11.5) and finally divide the sets w\+x in the order of their dimen¬
sionality as before.
Let u\+x (a») denote the image on £d* of w\+ 1 . In case w\+ x exists let u\+x (oO
denote the image on £a< of w2i+x, i = 0, • • • , k.
Let ak be a fc-cell of zpy and ak~i any one of its boundary cells. We see that on
£z*,
(11.6) w*+i(a*) — * dk bk + uk (ak~ i) (s = 1, or 2)
o fc_!
where there is one chain u*k{ak- 1) in the sum 2 corresponding to each (k — 1)-
cell ajk_i on the boundary of ak . Summing over all fc-cells of zp we find that
2) u*+l(Ot) zP + P”(zP) + 2/ S M*(a*-‘)-
ak ak a k„x
(11.7)
294
CLOSED EXTREMALS
[VIII]
To evaluate the double sum we invert the order of summation, and first consider
the sum
(11.8) 2 «;(«*-.)
° k
in which ak~ \ is fixed, and there is one term corresponding to each cell a k of zr
incident with ak~ 1. A pair of cells ( ak , ak~ i) corresponding to which ,s = 2 in
(1 1 .8) will be said to be of even type, otherwise of odd type. For pairs ( ak , ak~ i)
of even type,
£«*-i + (on IP)
by virtue of (11.5). Hence (11.8) reduces to a sum
taken for all pairs (a*, i) of even type with fixed and a a- incident with
a*-!. More generally we have
(11.9) 2 S =
ak « k - i
where the sum 2* contains a term iak~ i corresponding to each mutually incident
pair (<7a, at_i) of k- and (k — l)-cells on zv of even type.
I say finally that the sum
(11.10) 2*77/, = w*9
in which the terms are derived from those on the right of (11 .9), defines a (k — 1)-
cycle wp on IP. To see this we replace zv by a “simple” fc-cycle consisting of the
same fc-cells. The sums (119) and (11.10) will not thereby be altered. Upon
considering the circular sequence of k- and (k — l)-celJs then incident with a
given ( k — 2)-cell ak-2, k > 1, one sees that the number of pairs (aky a*_i) of
successive cells in this sequence which are of even type must itself be even.
This statement presupposes a sufficiently fine subdivision of zp. There are then
an even number of the cells ak~ i in (11.10) incident with ak-2 so that (11.10)
defines a ( k — l)-cycle. The cases k = 0 and 1 require no comment .
The homology (11.4) can accordingly be written in the form
(li.H) 2 «*+i -> ^ + w) + EwP
°k
where wp is a cycle of ( k — l)-cells of zv. The analysis of the homology (11.4)
is now complete.
We continue with the following theorem.
Theorem 11.1. Let S be a proper section of Rp belonging to g. Corresponding
to S and an arbitrarily small neighborhood N of a, there exists an arbitrarily small
[11]
THE NON-DEGENERATE CLOSED EXTREMAL
295
neighborhood No of a with the following property. Any k-cycle u on No {below c)
will be 12- homologous on N {below c) to a cycle on S .
Let r be any positive integer, and p the integer of the theorem. Let Srp
be a proper section of Rrp belonging to g. Let
M\ • , Mrp
be the successive manifolds on R which cut across g to define Srp. We suppose
that the manifolds
(11.12) M2r, • • - , MTp
are the manifolds which define the section 8 of the theorem.
If the respective components of No are sufficiently small, the statements of
the following paragraph will be true.
The components ur of the fc-cycle u of the theorem will satisfy the homologies
ur ~ vr [on N (below c)]
where vr is a ,7-normal A>cvcle. The cycle vr will possess a partition wrp on N
(below c). Hence
vr * wrp [on N (belowr c)].
The cycle wrp will possess an extremal projection xrp on 8rp such that
wTp * xrp [on N (below c))f
by virtue of the preceding lemma. Let yp be the extremal projection of xrp
on S, and zrp the r-fold partition of yp . Let (x) be an arbitrary point on xrpf
(7 r') the extremal projection of (71-) on S, and ( x ") the r-fold partition of (x').
The point (x") lies on zrp. Its vertices on the manifolds (11.12) of S are also
vertices of (x). We can accordingly use the deformation 77 of §7 to deform each
point (x) on xrp into the corresponding point (x") on zrp} holding the vertices on
S fast. We will then have the homology
xrp ^ zrp [on ]Sf (below c)].
But since zrp is the partition of yp we have
zrp * yv [on N (below c)].
Combining the preceding homologies we see that
uT * yp
[on N (below c)].
The theorem follows readily.
We continue with the following theorem.
Theorem 11.2. Let S be a proper section of Rf belonging to g. Corresponding to
S and to an arbitrary neighborhood N of a, there exists an arbitrarily small neigh¬
borhood N 0 of <r with the following property. If zp is any cycle on 8 such that
(11.13)
[on No {below c)],
zp * 0
296
CLOSED EXTREMALS
[VIII]
then
(11.14) zv ~ 0 [on S and Np ( below c)].
If the neighborhood No of the theorem is sufficiently small, the following state¬
ments are true.
The deformation dp(t) of §5 will deform the fc-cycle zv of the theorem into a
/-normal cycle wp such that
(11.15) wp * 0 [on N (below r)]
where the chains involved in (1 1.15) possess continuous extremal projections on
S which lie on Np (below c ). Suppose the extremal projections on S of the
(k + l)-chains involved in (11.15) sum to a (k + l)-chain fip. Under dp(t), zp
will be deformed through a (k -f- l)-chain which will possess a continuous ex¬
tremal projection bp on S. Moreover bp wall lie on Np (below c). Hence
bp +
and the theorem is proved.
The two preceding theorems lead at once to the following theorem.
Theorem 11.3. Let S be a proper section of Rp belonging to a non-degenerate
closed extremal g. Maximal sets of spannable and critical k-cycles on 0 belonging
to the critical set ar determined by g> can be taken as maximal sets of spannable and
critical cycles on S belonging to the f unction J(tt) on S and to the critical point (7r0)
on S determined by g.
The type number of g or a is accordingly the type number of (7 r0) as a critical
point of the function / defined by J(w) on S. Since / is non-degenerate, the tth
type number of / is then 5 * where k is the index of (7 r0) as a critical point of /.
Inasmuch as the index of g is by definition (see §10) the index of (7 r0) as a critical
point of /, we have the following corollary of the theorem.
Corollary. The ith type number of a non -degenerate critical extremal g of index
k is 8kif i — 0,1, • • • .
From Theorem 6.3 and the preceding corollary we deduce the following.
Theorem 11.4. If a and b are two ordinary values of J between which there is
just one critical value c taken on by just one non-degenerate closed extremal g , the
Q- connectivities of the domain J < b minus those of the domain J < a afford differ¬
ences which are all null except that
Case I: APk = 1,
or
Case II:
where k is the index of g.
A Pk-i = -1,
[12]
METRICS WITH ELEMENTARY ARCS
297
If Case I occurs, g is said to be of increasing type. Case I will occur if a span-
nable (k — 1 )-cycle is associated with g and this cycle bounds below c.
Metrics with elementary arcs
12. The space 12 and its 12-homologies depend upon the elementary extremals
used. Nevertheless we shall prove that the connectivities of 12 are topological
invariants of P, or, if one pleases, of the ra-circuit K of which R is the home-
omorph. To accomplish this we shall postulate the essential properties of
./-distances and elementary extremals in an abstract form. We shall thereby
define a metric with elementary arcs. Corresponding to each such metric a space
12 will be defined as previously. We shall then show that the connectivities of 12
are independent of the defining metric among metrics which are topologically
equivalent.
The basic elements form a set of “points” P, Q, R , etc. termed a “space” S.
To each ordered pair of points P and Q of the set S there is assigned a real number
PQ, satisfying the following postulates.
Postulate 1. PQ — 0 if arid only if P = Q.
Postulate 2.
(12.1) PR g PQ + RQ.
The number PQ is termed the distance from P to Q. See Lindenbaum [1],
Fr^chet [1].
These postulates are of the well known type used to define a “metric space.”
They are not sufficient for our purposes, but before proceeding further it will be
convenient to indicate several of their consequences. The first is as follows.
I. The distance PQ is never negative and, PQ = QP.
Upon setting R = P in (12.1) we see that PQ ^ 0. Upon setting P = Q in
(12.1) we then see that QR = RQ.
Before stating the second proposition let it be understood that a function/ of a
finite number of points Pl, • • • , Pn of S is continuous at a particular set Q1,
• • • , Qn of such points of S, if corresponding to an arbitrarily small positive
constant e , there exists a positive constant d so small that
l/(Pl, ,Pn) ~f(Q\ ,Qn) \<e
whenever
P<Qi < d (i = 1, • • • , n).
We now state the second proposition.
II. The distance PR is a uniformly continuous function of P and R as P and R
range over S.
This proposition follows readily from Postulate 2.
Let S be a second space possessing a metric satisfying Postulates 1 and 2.
Let P(Q) be a point on S uniquely determined by an arbitrary point Q on 2.
CLOSED EXTREMALS
298
[ VIII ]
The map P(Q) of X on S will be said to be continuous at a point Q0 on 2 if the
distance
P(Q)P(Qo) (on S)
is arbitrarily small when the distance QQo on 2 is sufficiently small. A home-
omorphism between S and 2 can now be defined in the usual way. In particular
a simple arc y on S will be defined as the homeomorph on S of the line segment
0 ^ g 1. If P, Q, R are three distinct points on y, Q will be said to be “be¬
tween” P and R on y if the image of Q on the t axis lies between the images of P
and R respectively.
We state two additional postulates concerning our space S.
Corresponding to 8 there shall exist a positive constant p, and corresponding to
each ordered pair of points P , R on 8 for which
(12.2) 0 < PR ^ p,
there shall exist at least one simple arc [PR] o?i S, with end points at P and R re¬
spectively, of such a nature that the following two postulates hold.
Postulate 3. If Q is any point on [PP] between P and R, then
(12.3) PR = PQ + QR .
Postulate 4. If Q is any point not on [PR], then
(12.4) PR < PQ + QR.
A simple arc satisfying Postulates 3 and 4 will be termed elementary. A space
S for which distances and elementary arcs can be defined so as to satisfy Postu¬
lates 1 to 4 will be said to possess a metric with elementary arcs. We shall denote
such a metric by Mp.
We continue with a number of propositions depending on our four postulates.
III. Corresponding to any pair of points P, R of 8 for which (12.2) holds, there
is but one elementary arc [PR] on S. Moreover as a set of points on S, [PR] = [RP].
Let [PR] be one elementary arc joining P to R. If there is any other such
elementary arc, let Q be any point on that arc between P and R. The point Q
must satisfy (12.3), and by virtue of (12.4) must then lie on [P/?].
One proves similarly that [PR] = [RP].
IV. Any two distinct points on an elementary arc [PR] bound an elementary arc
on [PR],
We shall first prove IV for the case in which one of the two points on [PII] is P,
and the other point a point Q between P and R.
Let W be any intermediate point on [PQ]. From (12.3) we have
(12.5)
PW +WQ = PQ.
METRICS WITH ELEMENTARY ARCS
299
[12]
Adding QR to both sides of (12.5), and applying Postulate 3 to the resulting right
member we see that
(12.6) pw + WQ + QR = PR.
Upon applying Postulate 2 to points W, Q, and R, and then to P, W, and R,
(12.4) takes the form
PW + WR = PR.
Hence W must lie on [PR],
Regard [PQ] as the homeomorph of a line segment 0 g t ^ 1, and let W(t)
be the point on [PQ] corresponding to t. Regard [PR] as the homeomorph of a
line segment Ogrg 1, and let <p(t) be the value of r at the point W(t). We see
that <p(t) is a continuous function of t. But the relation between the values of t
and <p(t) must be one-to-one. It follows that cp(t) is an increasing function of t.
Hence the point W of the preceding paragraph must lie on the segment of [PR]
bounded by P and Q.
Proposition IV is accordingly true if the given points are P and a point Q
interior to [PR]. Upon selecting a point V on [PQ], not P or Q , one proves
similarly that \l JQ] is an arc of [PQ] and hence of [PR].
Proposition I V is established.
V. On an elementary arc [PR], regarded as the homeomorph of a line segment
0 rg t S 1, the distance d(t) of the point t from the point t — 0 is a continuously
increasing f unction of t.
The continuity of d{t) follows from II. That d(t) increases with t now follows
from IV and Postulate 3.
The metric geometry which we have developed up to this point is closely
connected with the interesting and extensive geometry developed by Menger
[2, 3, 4]. Menger however starts with a different notion of “betweenness,” and
is not concerned with our problem in the large and function space 12.
The space S is said to be compact if each infinite set of points on has at least
one limit point on N.
VI. If S is compact , the point Q on an elementary arc [PR], at a distance s from
P, is a continuous point function Q(P , R , s) of P , R , and s, provided
(12.7) 0 g * g PR ^ P.
To prove this proposition let
Pn, Hn, S n (?) = 1, 2, • • •)
lx‘ an infinite sequence of points Pn and P„, and numbers s„, such that
0 ^ 8n ^ PnPn g P,
and such that
lim Pn = P, lim Rn = R, lim s„ = s.
300
CLOSED EXTREMALS
[VIII]
The points
Q(Pnj Rn , Sn) = Qn
will have at least one limit point on Sy say Q*. Without loss of generality we can
suppose that the sequence Qn has no limit point other than Q*, since such a
condition could be obtained by selecting a suitable subsequence of the points Qn -
From Postulate 3 we see that
(12.8) PnQn + QnRn = PnRn.
Letting n become infinite in (12.8), and using II we find that
(12.9) PQ* + Q*R = PR-
It now follows from (12.9) and Postulate 4 that Q* lies on [PR], But since
PQ * = lim PnQn = lim s„ = s,
n— *oo n— »oo
it follows from V that Q* is uniquely determined as the point Q(P, R, s). Hence
lim Q(Pn, Rn, s„) = Q(P, R, s).
n— »aO
Proposition VI is thereby proved.
Let S' and S " be two compact metric spaces provided respectively with
metrics M p> and M p*. We suppose S' and S" homeomorphic, and represent
corresponding points on S' and S" by the same letters P, Q, P, etc. We shall
avoid ambiguity by denoting the distance between points P and Q, on S' and S",
by the symbols
d'(P, Q), d"(P, Q),
respectively. In terms of these distances we state the following lemma. Its
form is convenient for future use.
Lemma 12.1. There exists a positive constant r' < p', such that two points P, Q
of S' and S" which satisfy the condition
(12.10) d'(P, Q) < r'
also satisfy the condition
(12.11) d"(P, Q) < — •
Similarly there exists a positive constant r" < p", such that two points P, Q of S'
and S" which satisfy the condition
(12.12) d'(P, Q) < r"
also satisfy the condition
(12.13) d\P, Q) <
[12]
METRICS WITH ELEMENTARY ARCS
301
To establish the existence of the constant r' we have merely to note that
d"(P, Q ) is a uniformly continuous function of P and Q on S', and that d"(P, Q) =
0 when d'(P, Q) = 0. The existence of the constant r " is similarly established.
The space tt determined by S. Let S be a compact metric space with metric
Mp. The distance between any two points P and Q on an elementary arc
[PQ] will be termed the Af-length of [PQ] or the M- distance between P and Q .
A set of p points on S, given in a circular order, two successive points of which
have an M -distance at most p, will be termed an admissible point (71-). We
suppose that p > 2. The space IIP corresponding to S will now be defined as
the totality of all admissible points (7 r) with p vertices, regarding two points (71-)
which can be obtained one from the other by a transformation of the group
Gp of §1 as identical. Contracted points on np are defined as previously.
With this understood the space 12 will be taken as the ensemble of the spaces
IF, p = 3, 4 • ■ ■ .
Chains and cycles on 12 are now defined as in §2. Replacing , /-length and
./-distance by M- length and M- distance respectively, partitions of chains on IF
are defined as in §2. Ordinary and special homologies on 12 are then introduced
as previously, leading finally to the definition of the connectivities of 12. It will
be convenient to term these connectivities of 12 the circular connectivities of S,
We now return to the spaces S' and S" and denote the corresponding spaces
12 by 12' and 12" respectively. We regard corresponding points on S' and S" as
identical. We note that points (tt) which are admissible on 12' or 12" may not be
admissible on 12" or 12' respectively. Points (t) or chains of points (t) which
are admissible both on 12' and 12" will ordinarily be denoted by the same symbol.
Let r0 be a positive constant such that
r0 < r', r0 < r",
where r' and r" are the constants of Lemma 12.1. Two points P, Q on S' and
S" will be said to be admissible rel r0 if
(12.14) d'(P,Q)<g, d"{P,Q)<\
A point (tt) will be said to be admissible rel ro if each pair of successive vertices of
(tt) satisfy (12.14). We now state the following lemma.
Lemma 12.2. Let z be a k-cycle on n'p and n"p which is admissible rel ro. If
z' and z" are respectively r-fold partitions of z on II 'rp and n"rp, then
(12.15) z* ~ z"
on both II'rp and II "rp.
We shall show that (12.15) holds on II 'rp. The proof that (12.15) holds on
II *rp is similar.
Let (tt) be a point on z , and P, Q a pair of successive vertices of (tt). Let h'
and h* be the elementary arcs determined by P, Q on S' and S" respectively.
302
CLOSED EXTREMALS
[ VIII ]
Let Q ' and Q" be points which divide h' and h" in the ratio of p to v on S' and S*
respectively. We suppose that p and v are positive integers such that
P -f- v = r.
The points Q' and Q" will be vertices on the r-fold partitions of (7r) relative to S'
and S" respectively.
We shall now prove that
Q") < P--
We start with the relation
(12.16) d'(Q', Q") ^ d'(Q'y P) + d'(P, Q").
With the aid of (12.14), and the fact that r{) < p we see that
dW, n ^ d'(Q, P) < ^° < p8'
Similarly
d'(P, Q ") ^ d"(Py Q) <
and from Lemma 12.1 we then infer that
d'(P, Q") <
From (12.16) we thus find that
dW, «'■) < I + 1 » ^
as stated.
The points Q' and Q ” can then be joined by an elementary arc k on S'. We
deform Q ' along k to Q ", moving Qf so that its distance on S' from its initial
position increases at a rate equal to the length of k on S'. The cycle z' will
thereby be deformed into the cycle z".
Let (tt') be the r-fold partition of (71-) relative to S' , and let (ir ,) be the point
through wThich (t') is deformed. It remains to show that (71- *) is admissible on
U,rT>y that is, that successive vertices of (irt) possess a distance at most p' on S'.
To that end let Qt be the point into which the vertex Q' is deformed at the time t.
We note that
d'(Qh P) ^ d'(Qt , Q') + d'(Q', P)
[12]
METRICS WITH ELEMENTARY ARCS
303
Thus the distance on S' between Qt and its adjacent vertices in (w t) will be at
most p'. The point ( irt ) is accordingly admissible relative to S'.
The proof of the lemma is now complete.
The principal theorem in this section is the following.
Theorem 12.1. The circular connectivities of two homeomorphic , compact,
metric spaces possessing elementary arcs are the same
As previously we denote the two spaces by S' and S" and represent corre¬
sponding chains on 12' and 12" by the same symbol.
To prove the theorem let H be any set of ft-cycles on 12' satisfying no proper
12 '-homology. Without loss of generality we can suppose that the cycles of H
are admissible rel r0, and consist of points (71- ) with a fixed number of vertices.
For if this were not the case, suitable partitions of the cycles of 11 would satisfy
these requirements and would again form a maximal set of A;-cycles on 12' subject
to no proper 12 '-homology.
So chosen, the cycles of H are cycles of 12" as well We shall now prove that
no proper combination z of ^-cycles of II satisfies an 12 "-homology
(12.17) 2*0 .
If the contrary were t rue there would exist a subdomain II " 9 of 12" upon which the
chains “involved” in the 12"-homology (12.17) would possess partitions. With¬
out loss of generality we can suppose q so large a positive integer that these
partitions lie on II'9 as well as on II "L The relation (12.17) thus implies an
homology
(12.18) 2" - 0
on both 11 "9 and II"', where z" is a partition of 2 relative to S".
Let z' be a partition on II'9 of z , relative to S'. It follows from Lemma 12.1
that
(12.19) 2' ~z"
on both II "9 and II'9. Moreover we have the special homology
(12.20) 2*2' (on 12').
From (12.18), (12.19), and (12.20) we infer that
2*0 (on 12'),
contrary to the nature of the set //.
We conclude that (12.17) is impossible, and that. the kth connectivity of 12" is
at least as great as that of 12'. Interchanging the roles of 12' and 12" we infer
that the kth connectivities of 12' and 12" are equal.
The proof of the theorem is now complete.
Returning now to our Riemannian manifold H we see that our elementary
extremals and /-lengths serve to define a particular metric on R with elementary
304
CLOSED EXTREMALS
[VIII]
arcs. We can thus regard R as a compact metric space S' with metric M'p<. If
the functional J is now replaced by another functional of the same general
character, R will give rise to a metric space S " with a different metric Mp-.
If points of S' and S" which represent the same point of R are regarded as corre¬
sponding, S' and S" are seen to be homeomorphic. We are thus led to the
following theorem.
Theorem 12.2. The circular connectivities of R are independent of a change in
the functional J with whose aid U is defined , provided J is replaced by a functional
of the same general character.
We add the following theorem.
Theorem 12.3. The circular connectivities of homeomorphic Riemannian mani¬
folds are equal. For such manifolds the circular connectivities never fail to exist.
By virtue of the preceding theorem any admissible function J can be used to
define the circular connectivities of R. In particular one can always take J as
the integral of arc length.
Moreover by virtue of the preceding abstract theory it is immaterial whether
the homeomorphism between two given Riemannian manifolds R' and R" can
be effected by analytic transformations or not, and this point is highly important.
It is sufficient that the compact metric spaces S' and S" respectively defined by
R' and R" and their geodesics be homeomorphic. This condition of home¬
omorphism between S' and S" is always fulfilled if R' and R" are homeomorphic.
CHAPTER IX
SOLUTION OF THE POINCARE CONTINUATION PROBLEM
The problem of the existence of closed geodesics on a convex surface was
considered by Poincare in connection with his studies in celestial mechanics
(Poincar<$ [2], Birkhoff [1, 3], Morse [7, 17], Schnirrelmann and Lusternik [1]).
Poincartf supposed that the given surface was a member of a family of convex
surfaces Sa depending analytically on a parameter a ranging over a finite
interval 0 g a ^ 1 . He supposed that one member of the family, say So, was
an ellipsoid with unequal axes. On the ellipsoid there are three principal
ellipses. According to Poincare, upon varying the parameter a, closed geodesics
appear and disappear in pairs and the analytic continuation of the three principal
ellipses will lead to an odd number of closed geodesics. One should recall that
there exist closed geodesics on So other than the principal ellipses, but if the
semi-axes of the ellipsoid are unequal and sufficiently near unity, the remaining
closed geodesics have lengths which are arbitrarily large. Certainly then for
values of a sufficiently near zero, an odd number of closed geodesics on Sa can
be obtained from the principal ellipses on So following the method of analytic
continuation.
Just how far this method can be carried is not clear. It is unquestionably
useful in limited cases. Relative to its general use the writer wishes to point
out certain difficulties and limitations.
Among the difficulties are the following. (1). The principle that closed
geodesics appear and disappear in pairs needs to take account of the fact that
infinite families of closed geodesics of the same length can appear on Sa for
isolated values of a when, for example, the surface becomes a sphere or a spheroid.
(2). As one varies a on the whole interval 0 ^ a ^ 1, it is conceivable that the
continuation of the principal ellipses may coalesce with the continuation of some
of the closed geodesics on So whose lengths were initially very large. In fact
one is really dealing with the continuation of an infinite class of closed geodesics
of infinitely many lengths, and not merely with a finite odd number of such
geodesics.
Limitations on the method are the following. (1). If one passes from the
2-dimensional to the m-dimensional ellipsoid, it appears that the number of
principal ellipses is sometimes odd and sometimes even, depending on m, and the
Poincar6 principle fails when the number is even. (2). The Poincare method
when valid affirms the existence of at least one closed geodesic, whereas in the
w-dimensional case we shall see that the existence of
m(m + 1)
2
305
306
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
closed geodesics with lengths commensurate in size with the lengths of the
principal ellipses on an m-ellipsoid can in general be affirmed to exist. (3). The
characterization of classes of closed geodesics according to the oddness or
evenness of the number of closed geodesics therein is obviously inadequate in
view of the possibility of classification by means of type numbers, and offers
no characterization for families of closed geodesics. (4). We shall see that it is
possible to give an adequate and general theorem on the continuation of the type
numbers of a critical set of closed geodesics, backed by existence theorems
applicable to each member of the family Sa. (5). The Poincar6 method does not
distinguish adequately between the continuation of the principal ellipses and the
continuation of the remaining geodesics on the ellipsoid. We shall show that it
is possible to relate the principal ellipses on the m-ellipsoid to a topologically
defined class of closed geodesics on any regular, analytic homeomorph of the
m-sphere. The geodesics so related to the principal ellipses on the m-ellipsoid
stand in remarkable metric, as well as topological, relations to these ellipses.
The existence of three closed geodesics on any closed convex surface subject
to certain limitations was first established by Birkhoff ([2], p. 180). Birkhoff
([3], p. 139) also established the existence of at least one closed geodesic on any
analytic homeomorph of an m-sphere.
The Poincare problem of the continuation of a closed geodesic on a convex
surface with respect to a parameter a will be solved as a part of a more general
problem which we formulate as follows.
Genkral Problem. I. To define 7 turner ical characteristics of sets of closed
geodesics on R) the possession of which by a particular set of closed geodesics is
sufficient to guarantee the existence of a corresponding set H of closed geodesics on
any other admissible Riemannian 7nanifold horneomorphic with R.
II. To show that the preceding set H varies analytically (in a manner to be made
precise) with any parameter a with which R varies analytically.
We shall solve this general problem, and apply our results to any Riemannian
manifold which is the homeomorph of an m-sphere. To that end we shall first
determine the circular connectivities of the m-sphere as defined in Ch. VIII.
We shall show in what sense the m(m + 1 )/2 principal ellipses on an m-ellipsoid
with unequal axes can be continued analytically, as the m-ellipsoid is varied
analytically. In the final section we shall state and prove a basic continuation
theorem.
Although we confine ourselves to the reversible case, the methods and results
hold with obvious changes in the irreversible case with positive integrand.
It seems appropriate in this place to point out the fundamental difference
between the methods employed by Birkhoff and those employed by the author
in the study of* periodic orbits. These methods are complementary. The
method most frequently employed by Birkhoff is based on the theory of fixed
points of transformations. Once a periodic orbit is given, the transformation
theory of Birkhoff is decidedly effective in discovering the infinitely many other
[1]
REGULAR SUBMANIFOLDS OF Rv
307
closed orbits which may exist nearby. The Birkhoff theory also affords a deep
characterization of local stability together with a development of the basic con¬
ceptions of recurrence and transitivity in the large. The reader is referred to
BirkhofTs numerous papers on this subject. The methods of the author are
generalizations for functionals of the theory of critical points of functions. The
theory of the author as developed so far has been successful in obtaining a priori
existence theorems on closed extremals, in classifying such extremals in the large,
and in solving the general continuation problem.
It is not implied that either theory is inapplicable to the domain of the other.
In fact for dynamical systems with two degrees of freedom the transformation
theory is highly successful both in the small and in the large. This is doubtless
due to the fact that the theory of the distribution of vectors in a 2-space is
essentially equivalent to the theory of critical points of a function of two vari¬
ables. In fact such a system of vectors, if suitably altered in length, will in
general become the gradients of a function. This relation between the fixed
point theory and the critical point theory does not persist however in higher
spaces.
On the other hand investigations now under way by the author as to the
manner in which the “index” of a closed extremal g is related to the index of
multiples of g indicate a closer connection between the formal aspects of the two
theories than has yet been disclosed. A preliminary paper by Hedlund [1]
in the case n = 2 bears this out. In all events the further study of the inter¬
relations between the two theories seems likely to be one of the most fascinating
subjects for research in the future.
The reader may also refer to a paper by Schnirrelmann and Lusternik [1].
In addition to refining the results of Birkhoff concerning the closed geodesics on
the homeomorph of a 2-sphere these authors introduce a deformation principle
which leads to certain types of critical points of functions and functionals.
Regular submanifolds of Rp
1. We continue the theory of non-degenerate closed extremals g of §11, Ch.
VIII. As previously let c, be the ./-length of g, and p any positive integer such
that pp > c. It (7T0) be a point on Rp which determines g. Suppose that the
length of none of the elementary extremals of g( t0) is 0 or p. Let the arc length
on g be measured in an arbitrary sense from an arbitrary point. Starting with
an arbitrary vertex of (7 r0) let the values of t at successive vertices of (w0) be
h <U < * • • < tp.
Let the local coordinates (x) on R neighboring the gth vertex of (to) be denoted
by ( zQ ), and neighboring this vertex suppose g has the representation
a-! = x ?(<) (q = 1, • • • , p; i = 1, • • • , m).
308
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
Let Mi, • • • , up be parameters which assume values near t\, • • • , (,, respectively.
The p-manifold
(i = 1, • • • , m),
(1.0) .
xpi = xPt(ur)
will be a regular submanifold of Rp neighboring (7r0). It will be termed the
extremal manifold on Rp neighboring (tt0).
Let Z be an arbitrary regular analytic submanifold of Rv passing through
(t0). We here admit only those regular submanifolds of Rv which consist of
points (t) none of whose elementary extremals have the length 0 or p. If the
manifold Z has no tangent line in common with the extremal manifold (1.0) at
the point (7r0), Z wall be termed a non-tangential submanifold of Rv belonging
to g. If Z is regularly represented in terms of parameters (v) in the form
A = <P9i(v) (q = L • • * , P; i = 1, * • * , m)
with (v) = (0) corresponding to (t0)} a necessary and sufficient condition that
Z be non-tangential is that when (v) = (0), the columns of the functional matrix
of the functions (pq{(v) be independent of the columns of the functional matrix of
the functions xq{(u) when ( u ) — (th • • ■ , ip).
We shall prove the following theorem.
Theorem 1.1. Corresponding to any non-tangential submanifold Z of Rp
belonging to g, there exists a proper section S of Rp belonging to gy possessing the
following properties. If (to) is the point on Z which determines g, the correspond¬
ence between points on Z sufficiently near (to) and their extremal projections on S
is one which is analytic and non-singular , and in which points (t) on Z can be
J-deformed into their extremal projections on S by suitably moving their vertices
along the corresponding curves g(T).
We shall confine the proof of the theorem to the case of orientable manifolds
R. On such a manifold the neighborhood of g can be referred to an analytic
coordinate system
(x, Vi, ■ ■ • , V«) (« = m - ] )
such that g corresponds to a segment of the x axis of length and two points
(xr y) whose coordinates ?/» are the same but whose coordinates x differ by an
integral multiple of oj, represent the same point on R. Such a coordinate
system will be explicitly exhibited in the case of the closed geodesics to wrhich wre
shall apply the theorem, so that any general proof of the existence of such a co¬
ordinate system can be omitted.
Let the #th vertex of (ro) be represented by the point x = aq on the x axis,
with
(LI)
a1 < a2 < ••• < aT < a1 + w.
[1]
REGULAR SUBMANIFOLDS OF R»
309
In terms of the parameters ( v ) representing Z the ?th vertex of (r) will have an
image in the space (x, y) such that
(1.2) x = x«(v) w = 1, • ' ' , V)
where
xq(fi) — aq>
and where the functions xq(v) are analytic in the variables ( v ) for (v) neighboring
(0). It will now be convenient to set
Xp+l(v) = Xl(v ) -f- (X).
The elementary extremal of g(n) which begins at the gth vertex of (t) will have an
image in the space (x, y) of the form
(1.3) yi = yqt(x, v) (i = 1, • • • , n)
for x on the interval
(1.4) xq(v) g x ^ xy4l(e) (q — 1, • • • , p)
where the functions yQi(x} v) are analytic in x and ( v ) for x on the interval (1.4)
and (v) neighboring (0).
We shall now determine the analytic consequences of the fact that Z is a non-
tangential submanifold of Rp. The r/th vertex of a point (t) on Z has a repre¬
sentation in the space (x, y) of the form
Vi = yqr(xq(v)> v)y
x — xq{y).
Since the x axis is an extremal, we see that
(1.5) y?,Or,0)s 0.
Let us indicate partial differentiation with respect to vh by adding the sub¬
script h. A necessary and sufficient condition that Z have no tangent line in com¬
mon with the extremal manifold (1.0) at the point (t0) is that the matrix
(1-6) || yh« 0) II (h = 1, • • • , r; i = 1, • • • , n; q = 1, • • • , p)
of qn rows and r columns have the rank r.
We are now in a position to choose the proper section S of Rv whose existence
is affirmed in the theorem. We take the manifold on R on which the ^th vertex
of the point (ir) on S rests as the image on R of the n-plane
0.7)
x = aq + e,
310
SOLUTION OF THE POINCARft CONTINUATION PROBLEM [ IX j
where e is a positive constant yet to be determined. The extremal projection
on S of the {joint (v) on Z will be a point with 9th vertex
(1.8)
Vi = yVi(aQ + P, »),
x = aq + e.
We choose the constant c so small that the functional matrix
(1.9) II ylkia" + «, 6) ||
has the rank r. This is possible since the matrix (1.6) Has the rank r.
From the fact that the matrix (1.9) has the rank r it follows that the relation
(1.8) between the point (v) on Z and the corresponding point on S is one-to-one,
non-singular, and analytic, provided (v) is sufficiently near (0). Finally any
point (tt) on Z (below c), sufficiently near (ir0) can be deformed (below c) into its
extremal projection (t') on S by moving the r/th vertex of (r) along the gth
elementary extremal of g(v) to the <7th vertex of (i r'), moving each vertex at a
J-rate equal to the J-length to be traversed.
The proof of the theorem is now complete.
We continue with the following lemma.
Lemma 1.1. Let f(v i, • • • , vk) be an analytic function of the variables ( v ) neigh¬
boring ( v ) — (0), assuming a proper , relative maximum c when (v) — (0). // c is a
sufficiently small positive constant , the closure of the domain
(1.10) f(v) ^ c - e
neighboring (v) = (0) will contain no critical points off other than ( v ) = (0), and
will have the locus
(1.11) f(v) = c - e
for its boundary neighboring (v) = (0). Moreover any k-cycle on (1.10), below c}
which is not homologous to zero on (1.10), below c, will be homologous on (1.10),
below cf to the ( k — 1 )-cycle (1.11).
The point (v) — (0) affords a proper maximum tof(v), and must accordingly
be an isolated critical point. The lemma then follows readily except at most for
the concluding statement. This final statement of the lemma is a consequence
of the results on maximizing critical sets in §7, Ch. VI.
We first note that the function <p = c — / is a neighborhood function cor¬
responding to / and the critical point (v) = (0), at least on the domain (1.10).
According to the results on maximizing critical sets in Ch. VI the type number
mk of (v) — (0) equals unity, so that there is a single spannable ( k — l)-cycle in a
maximal set of such cycles corr <p g e. But the cycle (1.11) is such a spannable
(k — l)-cycle, and any cycle on (1.10), below c, which is not homologous to zero,
below Cy is likewise a spannable (k — l)-cycle, so that the concluding statement
of the lemma is true.
REGULAR SUBMANIFOLDS OF Rp
311
The principal theorem of this section is the following.
Theorem 1.2. Let Z be a non-tangential submanifold of Rp belonging to a non-
degenerate closed extremal g. Suppose the index k of g is positive and equals the
dimension of Z. If J( w) assumes a proper maximum c on Z at the point (i r0)
which determines g> the locus
(1.12) JU) = c - e (on Z)
will be a spannable ( k — 1 )-cycle on 12 belonging to g, provided e is a sufficiently
small positive constant.
Let (v) be a set of parameters in a regular representation of Z neighboring
(tt0). Suppose that (v) — (0) corresponds to (7r0). l^et f(v) be the value of
J(tt) at the point (tt) determined by (i>). The function f(v) has a proper maxi¬
mum c at the origin (v) — (0). We now identify this function f(v) with the
function f(v) of the preceding lemma, and choose e so that, the lemma is satisfied.
Corresponding to Z let S be the proper section of Rp belonging to g whose
existence is affirmed in the preceding theorem. Let T denote the correspond¬
ence between a point (7 r) on Z and its extremal projection on S. We suppose
that the constant e in (1.10) is so small that the correspondence T between the
domain (1.10) on Z, and its extremal projection f on S will be analytic and
non-singular. If the extremal projection on f of a point (v) on (1.10) be assigned
the parameters (v), f appears as a regular, analytic, ^-dimensional submanifold
of *S.
Let F(v) denote the value of J(t) at the point on f determined by (v). We
shall continue with a proof of the following statement
(A). If 7] is a sufficiently small positive constant , the locus
(1.13) F(v) = r — 77 (on S )
neighboring (v) = (0) will be a spannable (k — 1 )-cyclc s*_i belonging to the critical
set <7 determined by g on 12.
To prove (A) first observe that
F(v) g /(f), F( 0) = m = c.
It thus appears that F(v) takes on a proper maximum c when (v) = (0). On the
other hand let (u) be a set of parameters in a regular representation of S neigh¬
boring the point (ti) which determines g. Suppose that (u) = (0) corresponds
to (7ri). Let \f/(u) be the value of J(r) at the point (tt) on S determined by (?/).
By virtue of the definition of the index and nullity of g in §10, Ch. VIII, ^(w)
will have in (u) = (0) a non-degenerate critical point of index k. Now f is a
regular, analytic submanifold of Sy and as such will be represented in the space
(u) by a regular, analytic sub-Axmanifold on which \p will assume a proper
maximum at the origin. It follows from Theorem 7.5 of Ch. VI that if 77 is a
sufficiently small positive constant, the locus (1.13) will be a spannable (k — 1)-
cycle on S belonging to the function \p(u) and the critical point (u) = (0).
312
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
But according to Theorem 11.3 of Ch. VIII such a spannable ( k — l)-cycle,
if sufficiently near (V i); will be a spannable ( k — l)-cycle on 0 belonging to the
critical set a determined by g.
Statement (A) is accordingly proved.
The cycle s*-! of (A) lies on S . Suppose that it is the extremal projection on
S of the cycle zk~ i on Z. As stated in the preceding theorem Zk-i can be
J-deformed on IP, into sk~i on S , arbitrarily near <rp, if sk~i is sufficiently near
<tp. If then 7} is sufficiently small, zk~ i will share with sk~ i the property of being
a spannable cycle on ft belonging to <r.
But Zk~ i lies on the domain of Z defined by (1.10), and being spannable
cannot bound on this domain below c. It follows from the preceding lemma that
zk~ i is homologous below c on the domain (1.10) of Z to the ( k — l)-cycle (1.11)
of Z. The (k l)-cycle (1.11) must then share with zk~ i and sk~i the property
of being a spannable ( k — 1 )-cyc!e on 0 belonging to the critical set a.
The proof of the theorem is now complete.
Geodesics on m-ellipsoids
2, We shall reduce the determination of the circular connectivities of the
m-sphere to an analysis of the closed geodesics on ellipsoids. We begin with an
ra-ellipsoid Em(ah • • • , am+i) given by the condition
(2.1) a\w\ + • * * + am+l^m+l =
where
(2.2) di > 0 (i = 1, • • • , m -f 1).
By the principal ellipse of Em(a), i 9^ j, we mean the ellipse in which the
2-plane of the Wi, Wj axes meets Em{a). The number of principal ellipses is
(m + l)m/2. These principal ellipses are closed geodesics. The determina¬
tion of the indices of the principal ellipses on Ern(a) can be reduced to a determi¬
nation of the indices of the principal ellipses of an ordinary ellipsoid E2(a).
We now investigate the principal ellipses on E2.
The ordinary ellipsoid E2(a). Let S represent a regular, analytic, orientable
surface S, and g a simple closed geodesic on S. Since S is orientable, g has two
sides, one of which may be termed positive and the other negative. Neighboring
g let S be referred to coordinates (x, y) of which y represents the geodesic distance
from a point P to g} taken as positive on the positive side of g and negative on
the negative side of g. Let x represent the distance along g in a prescribed sense
from a prescribed point on g to the foot of a geodesic through P and orthogonal
to g. If w is the length of gy each point P will possess infinitely many x-coor-
dinates, x + /zw, where m is an integer, positive, negative, or zero. It will
simplify matters if we think of the problem as given in the (x, y)- plane with the x
axis an extremal. * For our purposes the integral of arc length can be taken in
non-parametric form with x as the independent variable and y the dependent
[2]
GEODESICS ON r*-ELLIPSOIDS
313
variable. The integrand then has the period co in z, and the problem is of the
same nature as*the problem in §11, Ch. III.
As shown in Bolza [1], p. 231, the Jacobi equation corresponding to g or the x
axis takes the form
(2.3) g + K(x)y = 0,
where K(x ) is the total curvature of S at the point ion g. The nullity of g as a
closed extremal, determined in accordance with Theorem 10.1 of Ch. VIII, will
be the number of linearly independent solutions of (2.3) of period co. The
nullity of g is thus 0, 1, or 2. We continue with the following lemma.
Lemma 2.1. If the nullity of g is 1 and u(x) is a non-null solution of (2.3) with
period co, the only points x for which x is conjugate to x + are the points at which
u{x) = 0.
Let x — a be a point conjugate to x = a ~f co. Let v(x) be a solution of
(2.3) such that
v(a) = 0, v'(a) ^ 0.
We make use of Abel's integral, by virtue of which
(2.4) u(x)v'(x) — u'(x)v(x) = const.
By hypothesis
v(a + co) = 0, v'(a -f co) ^ 0.
Upon setting x — a and then a -f co in (2.4), we find that
(2.5) u(a)[v'(a) — v'{a -f co)] = 0.
Condition (2.5) leads to two cases.
Case I. u(a) = 0.
Case II. u(a ) 5* 0.
If Case II holds, it follows from (2.5) that
v'(a) = v'(a + co),
and since
v(a) = v(a -f- co) = 0,
we conclude that v(x) has the period co. But v(x) is independent of u(x) since
in Case II
v(a) = 0, u(a) t* 0,
from which it follows that the nullity of g is 2. From this contradiction we infer
that Case II is impossible.
314 SOLUTION OF THE POINCARE CONTINUATION PPOBLEM [ IX ]
Thus Case I holds and the lemma is proved.
Let 5 denote the arc length along the ellipse 01;. Let the length of 0*, be de¬
noted by gi}. We shall prove the following lemma.
Lemma 2.2. Corresponding to constants ai > a2 > a3 sufficiently near unity ,
the principal ellipses of E2(ah a2, a3) have the following properties.
(a) . To each point s 0 on gX2 there corresponds just one conjugate point s for which
So < s < So + g i2 while So is never conjugate to So -f- g 12 -
(b) . To each point s0 on g 23 there correspond just two conjugate points x for which
So < s < So + g 23 while s0 is never conjugate to s0 + 023*
(c) . On g 13 opposite umbilical points are conjugate to each other and to no other
points. The geodesic gX3 is non-degenerate.
We need the fact that the total curvature K(s) of E2(au a2, a3) at. the point s
on gi2 will increase with a3. lo see this one notes that K(s) is the product of the
curvature k 1 of g12 at the point s, and the curvature k2 of the ellipse in which a
plane orthogonal to gl2 at the point s cuts E2(ah a2, a3). An increase of a3 will
not alter kh but will diminish the axis of the ellipse orthogonal to g12. It will
accordingly increase k2l and hence K(s).
We shall prove the following statement. In this statement we set the length
g 12 =
(A). On the ellipse gl2 of E2(aly a2, a2) the distance As from a pomt s to its first
follmving conjugate point, measured in the sense of increasing s, exceeds co/2 for all
points s except the intersections of gX2 with the W\ axis, for which points As — co/2.
Let s be measured from the point of intersection of gX2 with the positive w 3
axis. All geodesics through the point s = 0 on gn on E2(a}, a2, a2) go through
the opposite point on gV2, and form a field otherwise. These geodesics are
ellipses. It follows that the point s = 0 is conjugate to the points s = co/2
and s = won 0i2, and that any solution of (2.3) which vanishes when x = s = 0
has the period co.
On the other hand on the ellipse gX2 of E2(ah a2, a2)} K(s) is less than the total
curvature at the same point on E2(alf a2, cu). But on E2(ah a2 , ax) the point
5 = u/ 4 on gi2 is conjugate to the opposite point on gi2 and to no other points on
012, as one proves by considering the geodesics through s = co/4 on gX2. An
application of the Sturm Comparison Theorem to (2.3) now' shows that the dis¬
tance from the point s = co/4 on gi2 to its first conjugate point on gi2 exceeds
co/2, on E2(ah a2f a2). It follows from Lemma 2.1 that the same is true for all
point0 of 012 on E2(ah a2y a2) except the points conjugate to s = 0. To see this
one varies the initial point s from co/4 to 0 or to co/2. If during this variation the
distance As from s to its first conjugate point should reduce to co/2, the symmetry
of the ellipsoid shows that the point s would be conjugate to the point s + co,
contrary to Lemma 2.1, unless s = 0 or co/2.
Thus statement (A) is proved.
We can now prove (a) of Lemma 2.2.
The curvature K(s) on the ellipse gX2 of E2(ah a2 , a3) is less than that at the
[2]
GEODESICS ON m-ELUPSOIDS
315
same point on E2(ah a2, a2). It follows from (A) and the Sturm Comparison
Theorem, that to each point s on the ellipse gn of E2(ah a2, a3) there corresponds
at most one conjugate point prior to or including s + to. If we compare E2(ah
a2, a3) with the unit sphere, we see that if the constants a* are sufficiently near
unity, there will be exactly one conjugate point of the point s prior to the point
s + won and the point s + to will not be conjugate to s. The proof of (a)
is now complete.
To prove (b) one first proves the following statement, setting the length 023 = to.
(B). On the ellipse g23 of E2(a2j a2) a3) the distance A s from a point s to the
first following conjugate point is less than co/2 for all points except the intersections
of <723 with the Wz-axis, for which points As — to/2 (.023 — to).
To prove (B) we compare 023 on E2(a2y a2, as) with g23 on E2(a 3, a2 , a3), and
use Lemma 2.1 as in the proof of (A). To prove (b) we make use of (B), com¬
paring 023 on E2(a 1, a2, a3) with 028 on E2(a2) a2, o3).
To prove (c) we recall that the geodesics through an umbilical point pass
through the opposite umbilical point, but form a field otherwise. Hence each
umbilical point on gX2 is conjugate to the opposite umbilical point, and to no
other points on 0i2.
We shall now prove that the nullity of 0u is not 2. If the nullity of 013 were 2,
each point s on 0i3 would be conjugate to the corresponding point s + to. After a
slight decrease of a2 on E2(ah a2, a3), no point $ 4 to would be conjugate to the
corresponding point s, contrary to the fact that there would still be umbilical
points on gu if ax > a2 > a3. Thus the nullity of 0i3 cannot be 2.
Finally the nullity of 0i3 cannot be 1. If the nullity of 013 were 1, set s = x
and let u(x) be the non-null periodic solution of (2.3). According to Lemma
2.1, u(x) would vanish at each umbilical point, since each umbilical point s0 is
conjugate to the point s0 4 u. Thus u(x) would vanish at each of the four
umbilical points, contrary to the fact that these four points are not mutually
conjugate.
Thus the nullity of 0i3 must be zero, and statement (c) is proved.
We shall conclude this section with a proof of the following theorem.
Theorem 2.1. If the constants ax > a2 > a3 of the ellipsoid
(2.6) a\w\ + a\w\ + a\w\ = 1
are sufficiently near 1, the principal ellipses 0i2, g 13, and g23 are non-degenerate and
possess the indices 1, 2 and 3 respectively.
Let distances s on 0i3 be measured from an umbilical point. By virtue of (c)
in Lemma 2.2, there is just one point on 0i3 conjugate to the point s = 0 on the
interval 0 < s < w, where w = 013. Moreover the point s = 0 is conjugate to
the point s = a>. It follows from (A) in §11, Ch. Ill, that the index of 0i3 is 2.
It follows similarly from (a) and (c) in Lemma 2.2, and from (B) in §11, Ch.
Ill, that the indices of gi2 and gn are 1 and 3 respectively if the semi-axes of
(2.6) are sufficiently near unity.
316
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
In Lemma 2.2 (c) we have expressly affirmed that gu is non-degenerate. That
012 and 023 are non-degenerate if the semi-axes of E2(a) are sufficiently near unity
follows respectively from the statement in Lemma 2.2 (a) that on 0J2 a point s is
never conjugate to s + 0i2, and the statement in (b) that on a point s is
never conjugate to s + g2s-
The proof of the theorem is now complete.
The indices of the ellipses 0,;
3. We shall determine the indices of the principal ellipses 0i7 of the ra-ellipsoid
Em(a) of (2.1). By the principal ellipsoids of Em(a) we mean those 2-dimen-
sional ellipsoids which are obtained from Em(a) by setting all of the coordinates
(w) equal to zero save three.
The equations of the geodesics on Em(a) can be put in the form
(3.1) ' w] + X Wj-a) = 0 (j not summed),
(3.1) " a\w\ = 1 (i,j = 1, • • • , m + 1),
where the independent variable is the arc length s, and (w) and X are dependent
variables to be determined as analytic functions of s by the conditions (3.1).
It follows from the equations (3.1) that the geodesics on any principal ellipsoid
of Em(a) are geodesics on Em(a).
We shall prove the following theorem.
Theorem 3.1. The index and nullity of the principal ellipse gii on the ellipsoid
Em(a) is the sum of the indices and nullities of 0ty regarded as an ellipse on each of
the m — 1 principal 2-dimensional ellipsoids on which it lies.
For simplicity we shall give the proof of this theorem for the ellipse gm, m+ 1.
We denote this ellipse by g.
We shall need a parametric representation of Em(a) neighboring g. Such a
representation can be given in terms of parameters
Or, 2/1 , ,yn) (n = m - 1)
with the points (y) = (0) corresponding to g . In fact if we set
r%) = 1 - alyl (h = 1, • • • , n),
we can represent Em{a) near g parametrically in the form
wa = yh (h = 1, • • • , n),
(3.2)
(r > 0),
[3]
THE INDICES OF THE ELLIPSES gif
317
The variables ( y ) are limited to sets near (0). If the representation is to be one-
to-one, it will be necessary to limit x to some such interval as the interval
0 < x ^ 2tt.
As in §10, Ch. VIII, we associate g with a boundary problem in the space
(xy y) in which the integral is the arc length on Em(a)f and the end conditions
require that x = 0 at the first end point, x = a? at the second end point, and
that at these end points the coordinates y * be the same. As we have seen in
§10, Ch. VIII, the index of g will be that of an index form corresponding to the
critical extremal
(y) = (0) (0 g * g 2*)
in this boundary problem. We proceed to set up such an index form.
We cut across the x axis by the four n-planes
(3.3) x = 0, x = L x = tv, x =
Z ju
Let
(3.4) P\ P\ P3, P4
be points on the respective n-planes (3.3) neighboring the x axis, and let
(3.5) 0 yqu • • • , Vi) (? = 1, • • • , 4)
be the ^-coordinates of the point Pq . Let
(3.6) (vXy • * • , vi) (6 = 4n)
denote the ensemble of the coordinates (3.5). Let the images on Em(a) of the
points (3.4) be joined in circular order by minimizing geodesics, and let the sum
of the lengths of these geodesics be denoted by J(v). The required index form
will be the form
(3.7) Q(v) = Jvav0( 0)vave (a,0 = 1, • • • , 8).
It follows from the symmetry of the ellipsoid Em(a) with respect to the ?n-plane
wk = 0, that the function J{v) and the form Q(v) will be unchanged in value if
for a fixed k we replace
(3.8) yl, y\ , y\, yt (fc = 1, • • • , n)
by
2/jfcJ ykl Vkl Vk’
It follows that Q(v) can contain terms which are constant multiples of the
product
yivi
(i,j = 1, • • • ,4 ;h,k = 1, • • • ,n)
318
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
only if h = k. We can accordingly write Q( v) as a sum
(3.8)' Q(v) s Qi(y\, y\, y\y y\) + * * * + Qn(yl> yl, yl, yl)
where Qk is a quadratic form in the arguments (3.8).
Hence the index of Q(v) will be the sum of the indices of the separate forms
Qk . But if we set all the variables ( v ) equal to zero save those in Qkf we see that
Qk is an index form which would be associated with g , were g regarded as a closed
geodesic on the principal ellipsoid which lies in the space of the
wky wn, wm+ 1 (k = 1, • • • , m — 1)
axes.
Similar results hold relative to the nullity of Q(v), and the proof of the theorem
is complete for the case of gm , m+i- The analysis is not essentially different for
the case of gxj in general, and will be omitted.
We can use the preceding theorem and Theorem 2.1 to establish the following
theorem.
Theorem 3.2. Corresponding to constants a i > • * • > am+i sufficiently near
unity , the principal ellipses gxj of the m-ellipsoid Em(a) will be non-degenerate and
possess indices k given by the formula
k = m-\-i+j — 4.
We suppose that- i < j. Let p, gy and h be three distinct integers on the
range 1, • • • , rn + 1. Let
Epqh
be the principal ellipsoid which lies in the 3-space of the wVJ wq} wh axes. In
particular consider the principal ellipsoids Eijh which contain gXJ. There will
be (i — 1) such ellipsoids for which h < i. On each such ellipsoid, gxj will be
of the type of g2z in Theorem 2.1, and have the index 3, provided always the
constants ai> • • • > Om+i are sufficiently near unity. There will be (j — i — 1)
ellipsoids Eijh for which h lies between i and j. On such ellipsoids, gXj will be
of the type of gn in Theorem 2.1, and will have the index 2. There will be
(m — j + 1) ellipsoids Eijh for which h > j. On such ellipsoids, gxj will be of the
type of 012 in Theorem 2.1, and will have the index 1. By virtue of the preceding
theorem the index of gxj on Em(a ) will be the sum of these indices. Thus
k = 3 (i - 1) -f 2(j - i - 1) + (m - j + 1) = m + i + j - 4
as stated.
That ga will be non-degenerate if the constants ai > • • • > am + i are suffi¬
ciently near unity is similarly proved.
The geodesics 0jy(a). Let the closed geodesic which covers gxj r times on
Em{a) be denoted by 0t-,(a). The preceding theorem will now be extended to
include an evaluation of the indices of the geodesics gr%i(a).
[4]
THE EXCLUSIVENESS OF THE CLOSED GEODESICS g\i
319
Theorem 3.3. Corresponding to constants ai > • - • > am+ 1 sufficiently near
unity , the geodesics gTij(a) for which r is less than a fixed integer s mil be non-degen¬
erate and possess indices &J7 given by the formula
(3.9) kij = m + i + j — 4 + 2 (r — 1 )(m — 1).
To prove this theorem we review the proof of Theorem 2.1 and verify the truth
of the following extension. If corresponding to the integer r the constants
ai > a2 > a3 of the ellipsoid E2{a) of Theorem 2.1 are sufficiently near unity,
the geodesics
012(a), 013(a), 023(a)
will be non-degenerate and possess indices given respectively by the formulae
2 (r - 1) + 1, 2 (r - 1) + 2, 2(r - 1) + 3.
The proof of this statement depends upon an obvious restatement of Lemma
2.2.
We next review Theorem 3.1, and verify the fact that the index and nullity of
grij(a) are the respective sums of the indices and nullities of g J;(a ) on each of the
(m — 1) principal 2-dimensional ellipsoids on which g'jia) lies. We then reason
as in the proof of Theorem 3.2, and conclude that <7*7(a) is non-degenerate, and
that its index k is given by the formula
*1/ - [2(r - 1) + 3] (i - 1) + [2(r - 1) + 2] (j - t - 1)
+ [2(r - 1) + 1] (m -j + 1),
provided the constants ai > * • - > am+i are sufficiently near unity.
This evaluation 6f k -7 reduces to that given in (3.9), and the proof of the
theorem is complete.
We verify the important property that for arbitrary integers i < j on the range
1, * * * , m + 1,
hr < Jr r+ 1
Kij ^ K\2 >
and that in particular for any positive integer r,
L r _ Lr+l
Km, m+ 1 — ^12 •
The exclusiveness of the closed geodesics g -7-
4. In this section we are concerned again with the m-ellipsoid
(4.1) = 1 (i = 1, •••,?«+ 1)
where a» > 0. We shall make use of the equations of the geodesics on (4.1)
in the form (3.1). If Wi = Wi(s) represents a geodesic on (4.1), the function
\(s) in (3.1) can be determined as follows. Two differentiations of (4.1) with
respect to s yield the identity
a2{w'tw't + a^Wxw" = 0
(i — 1 , ■ • * . m -j- l )
320
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX I
Upon substituting — \a\vx for w\ in accordance with (3.1) with i not summed,
we find that
(4.2)
2 ' '
ajWjWi
a\wiWx
But since s is the arc length, we have
(4.3) w'w'== l.
When the constants at in (4.1) are all unity, we see from (4.2) that A(s) = 1.
Accordingly, for constants cu sufficiently near unity, X will be uniformly near 1
for any point ( w ) on the ellipsoid (4.1) and set (w') satisfying (4.3).
Before coming to the theorem it will be convenient to state a lemma.
Lemma 4.1. Let <p(s) be a function which is continuous in s and has the period a>.
If u(s ) is a solution (f^O) of the differential equation
(4.4) w" -f <p(s)w — 0,
of period w, such that all solutions of (4.4) of period w are dependent on u(s ), the
only solutions of (4.1) whose zeros s have the period w are dependent on u(s).
The proof of this lemma is similar to that of Lemma 2.1 and will be omitted.
The principal theorem of this section is the following.
Theorem 4.1. Let N be an arbitrarily large positive constant . Corresponding to
constants a\ > • • • >0^+1 sufficiently near unity , there are no closed geodesics on
Em(a) with lengths less than N other than the geodesics gri3(a).
We begin by stating limitations on the constants o* under which we can prove
the theorem is true.
Corresponding to any set (a) of m + 1 positive constants, let g be a geodesic
on the ra-ellipsoid Em(a), and \0(s) the value of X wrhich with g satisfies the
system (3.1). Let c be any number, and Wj(s, c) a solution of the differential
equation
(4.5) ' w* + \{s)a)w = 0
such that
(4.5) " Wj(c, c) = 0, Wj(c, c) = 1,
where j is one of the integers 1, ■ • • , m + 1.
Upon referring to (4.2) we see that \0(s) s 1 when (a) = (1), independently
of the choice of the geodesic g on (4.1). For (a) = (1) the function Wj(s, c) will
then vanish at the points 5 = c + mr, where n is an integer, positive, negative,
or zero. For (a) = (1) the qth. zero of Wj(sy c) following s = c will thus lie on
the interval
(4.6)
c + (2# — l)—<s<c + (2<? + 1) -g .
[4]
THE EXCLUSIVENESS OF THE CLOSED GrEODESICS g\ .
321
Let At be a positive integer such that the constant N of the theorem satisfies
the condition
N < (2m + D~
Let q be a positive integer at most n. We restrict the constants (a) to so small a
neighborhood K of the set (a) = (1) that corresponding to any geodesic g on
Em{a)1 the gth zero s > c of the function wfs, c) lies on the interval (4.6) and is
the only zero of Wj(s, c) on the closure of that interval. Such a neighborhood K
can be chosen independently of the choice of c, or of the geodesic g on Em(a), or
of the integer.; in (4.5)'.
We shall now prove the following statement.
(a). Corresponding to any set (a) which lies on the preceding neighborhood K of
the set (a) = (1) and satisfies the conditions
(4.7) fll > • • • > Um + l > 0,
any closed geodesic g on Em(a) with a length < N will be a geodesic glfia).
Let g be represented in the form
(4.8) Wi = ut(s) (i = 1, • • • , m + 1).
Of the functions Ui(s) at least two functions, say
uh(s ), uk(s) (h < k),
are not identically zero. Let A(s) be the function with which g satisfies (3.1).
The functions uh(s) and uk(s) are solutions of the differential equations
(4.9) w'f + A (s)a\w = 0,
(4.10) w* + A(s)a^ = 0,
respectively. They have the period w in 5. We continue with a proof of the
following statement.
(f$). All solutions of (4.10) of period a> are linearly dependent on uk(s ).
Suppose (0) false. Then all solutions of (4.10) have the period a>.
The solution uh(s) of (4.9) vanishes at least once, since A ($) > 0. Suppose
that Uh(c) = 0. According to our choice of the neighborhood K , the gth zero,
0 < q ^ m, of uh(s) following s — c is on the interval (4.6), and is the only zero
of Uh(s) on the closure of that interval. Since Uh(c + w) = 0, and c o < N, the
point s = c co must lie on one of the intervals (4.6), say the pth.
Let Wk(s) be a solution of (4.10) such that
wk(c) = 0, w'k(c) = 1.
As previously noted wk(s ) has the period w if (/3) is false. Hence s = c + o> is a
zero of Wk(s). Inasmuch as wk(s) has just one zero on each of the intervals (4.6),
322
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
the point s = c + a? must be the pth zero following s = c, not only of uh(s ) but
also of wk(s).
Upon applying the well known Sturm Comparison Theorem to the differential
equations (4.9) and (4.10) and their solutions Uh(s ) and wk(s) respectively, we
see that the pth zero of uh(s) and wk(s) following $ = c cannot be common to
uh(s) and wk(s). From this contradiction we infer the truth of statement (0).
Now let Wj(s , a),j = 1, ••• , m + 1, be a solution of the differential equation
w* -|- \(s)a)w — 0,
such that
Wj(a, a) — 0, w '}(a, a) — 1.
Let Dj(a) be the distance along the 5 axis from s — a to the pth zero of Wj(s , a),
choosing p as previously so that
(2 p ~ 1) j < oj < (2p + 1)
We shall now prove statement (7).
(7). The function Dk(s) — a> is positive except at the zeros of uk(s ).
It is clear that Dk(s ) = a) at the zeros of uk(s) by virtue of our choice of p.
According to (/?), all solutions of (4.10) of period u> are dependent on uk(s)f and
according to Lemma 4.1, the zeros of uk($) are then the only values of s at which
Dk(s) — oj. Between each pair of successive zeros of uk(s), Dk(s) — a> must have
one sign. To determine that sign we compare uh(s ) with uk(s)f recalling that ak
in (4.10) is less than ah in (4.9). From the Sturm Theorems we can then infer
that there is at least one zero of uh(s ) between each two consecutive zeros of
uk(s). Suppose that uh(a) = 0. I say that
(4.11) Dk(a ) > a?.
To establish (4.1 1) we compare Uh(a) with wk(s, a), recalling that
uh{a) = wk(a, a) = 0.
According to the Sturm Comparison Theorem the pth zero of wk(sf a) following
s = a, follows the corresponding zero of Uh(a), namely 5 = a + a?, so that (4.11)
holds as stated, and (7) is proved.
One can similarly prove the following.
(6). The function Dk(s) — a? is negative except at the zeros of Uh(s).
We now complete the proof of (a).
Let Ui(s ) be any one of the functions Ui($ ) in (4.8), other than uh(s ) and uk(s).
I say that Ui(s) 0. For if h < k < i and u»(s) ^ 0, we could treat k and i as
we have just treated h and fc, and infer that Dk(s) < a> except at the zeros of
tt*(s), contrary to (7). If the integers h> k , i are in some order other than the
order h < k < i, we can interchange their r61es and arrive at a similar contra¬
diction. Hence Ui(s ) 35 0 for i not h or k. Thus g must be one of the geodesics grhk.
The proof of (a) and of the theorem is now complete.
15]
THE LINKING CYCLES A jr 2 (a)
323
The linking cycles A[2(a)
5. We shall eventually determine the circular connectivities of the m-sphere as
defined at the end of Ch. VIII. To that end we shall prove that the geodesics
g ^(a) for which r is at most a prescribed positive integer s will be non-degenerate
and possess linking cycles, provided the constants o% are sufficiently near unity
and satisfy the conditions
(5.1) ax > • • > Om+i.
We shall explicitly exhibit these linking cycles. We begin with the geodesics
01 2 (a)*
The cycles A [ 2 (a). Recall that Em(a) reduces to the unit m-sphere
(5.2) wtv)i = 1 (i = 1, • • • , m -f- 1),
when the constants a* = 1. We denote Em(l) by Sm.
Let A1 and A2 respectively denote the points of intersection of Sm with the
positive Wi and w2 axes, and let A be the point diametrically opposite to A1 on S.
Corresponding to g[2{ 1) we introduce a set of 2r — 1 constants eq, such that
(5.3) 0 < ex < • • • < e2r-i < 1.
In the space (w) let
(5.4) Mq (q = 1, ••• ,2r - 1)
be an (m — l)-sphere formed by the intersection of Sm with an m-sphere S„ of
radius eq, with center at A1 if q is even, and with center at A if q is odd. Let
(5.5) P\ • ■ ,PP (p = 2r + 1)
be a circular sequence of 2r + 1 points on Sm of which
(5.6) P1 - A\ Pp = A2}
and of which the points
(5.7) P2, • • • , P2r
lie on the respective (m — l)-spheres
(5.8) M\ • • , M2r~l.
Our integral on Sm is the arc length. The constant limiting the lengths of the
corresponding elementary extremals can be taken as any positive constant less
than 7r. If we choose p less than tt but differing from i r sufficiently little, any
two successive points in the circular sequence (5.5) can be joined by an elemen¬
tary extremal on Sm, and the points (5.5) will be the vertices of a point (7r) interior
to the corresponding space IP.
324
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
The ensemble of the points (7r) as the vertices (5.7) range over their respective
(m — l)-spheres (5.8) is a fc-cycle of dimension
(5.9) k = (2r — 1 )(m — 1).
We take this fc-cycle as our definition of A[2(l).
We shall take the constant p limiting elementary geodesics on Em(a) as the
constant p chosen above for Sm = Em( 1). Such a choice is permissible for
constants a* sufficiently near unity. We shall denote the spaces n*, R9, and 12
then determined by the integral of arc length on Em(a) by
IU(a), Rq(a)y 12(a),
respectively.
Let (t) be an inner point of ng(l). The vertices of (t) lie on the m-sphere
Em( 1). If the constants a, are sufficiently near unity, the central projections
of the vertices of (71-) on Em(a) will define a point (t') on n°(a). We term ( ir' )
the central projection of (tt) on IU(a). If the constants a* are sufficiently near
unity, the cycle A[2(l) will have a central projection on n2r+1(a), and we take
this central projection of A[2(l) as our definition of A[2(a).
We shall prove the following lemma.
Lemma 5.1. Corresponding to a prescribed positive integer r, constants ax > ••• >
am + 1 can be chosen so near unity that the cycle A [ 2 (a) will be a linking cych on
12(a) belonging to the critical set a determined by g\2 (a).
The lemma will follow after we have verified the truth of statements (a),
(0), and (7) below.
(a). The dimensionality k of the cycle A [2 (a), as given by (5.9), equals the index
k\2, as given by (3 .9) .
We have merely to set i = 1 andj = 2 in the index formula
Ki = m + i+ j- 4 + 2(r- 1 )(m - 1),
to obtain (5.9) as stated.
0). On A[ 2 (1), J{rr) assumes a proper absolute maximum at the point (7r0) which
determines p[2 0 )■ In terms of k\2 parameters (v) regularly representing A[2(l)
neighboring (xo), J{ 7t) has a non-degenerate critical point of index k[2 at the point
(i>0) corresponding to (tt0).
The first statement under (£) is a consequence of statement (A) in the proof of
Lemma 15.1 of Ch. VII. We refer to the case where the number of vertices,
p + 2, in the cycle Xp(m_i) of (A) equals the number of vertices, 2r + 1, in
the present cycle A[ 2 (1), that is the case where
p = 2r ~ 1,
Upon adding the elementary geodesic A1 A2 to the geodesic in (A) of §15,
Ch. VII, one obtains the closed geodesic g f2(l). More generally the addition
of the elementary geodesic A1 A2 to the broken geodesics of XP(m-i) of (A) yields
[5]
THE LINKING CYCLES Ar12 (a)
325
the respective broken geodesics g( tt) determined by points (7 r) on A[2(l)- The
first statement in (/S) is thus a consequence of (A) in §15, Ch. VII.
Let f(v) be the value of J (tt) at the point (7 r) on A [2(1) determined by the param¬
eters ( v ). The function /( v) forms an index function belonging to 02r-i, re¬
garding 02 r- 1 as a critical extremal of the boundary problem in which the func¬
tional is the arc length on Sm and the end points are fixed at A 1 and A2. Since
A1 and A2 are not conjugate on 02r-i, the point (v) = (v0) is a non-degenerate
critical point of f(v). Since f(v) assumes a maximum at (u0), the critical point
(t>o) must have an index equal to the number of parameters (v), namely k[2 .
Thus (/?) is proved.
(7). On A[2(a), J(tt) assumes a proper , absolute maximum at the point (iri)
which determines g[2{o), provided the constants a* are sufficiently near unity.
To represent A[2(a) neighboring (71-1), we make use of the preceding repre¬
sentation of A [ 2 ( 1 ) neighboring (71- 0) in terms of the parameters (i>) . We represent
points on A [ 2 (a) and A [ 2 (1 ) which are central projections of one another by the
same parameters (t>). Let/(t>, a) then denote the value of J(ir) at the point (w)
on A[2(a) determined by ( v ). According to (£0 the function f(v, 1) has a non¬
degenerate critical point of index k[2 when (v) — ( Vo ). Hence for constants a,
sufficiently near unity, f{v) a) will have a unique , non-degenerate critical point of
index k[2 neighboring ( v0 ). The coordinates ( v ) of this critical point will be
analytic functions \v{a) ] of the variables a*. But 0[2(1) projects centrally into
g [ 2 (a) . We i nf er that
M«)] — (^o)-
Let e be a positive constant so small that the domain of points ( v ) which satisfy
the condition
(5.10) f(v, 1) ^ f(v 0, 1) - e
and are connected to (v0) in the space (t>) contains no critical points of f(v, 1),
other than (v0). We now place two restrictions on the constants a%.
Let II be the closure of the set of points (t) on A [ 2 (a) which are not represented
by points (t>) on the domain (5.10). The first restriction on the constants at is
that they be so near 1 that the value of J(tt) at points (tt) on H is everywhere less
than/(i>0, o). The second restriction on the constants a, is that they be so near 1
that/(t>, a) has no critical points on the domain (5.10) other than the point (t>0).
By virtue of these two restrictions on the constants aiy J (t) assumes its absolute
maximum on A[2(«) at the point (717) corresponding to (v0), that is, at the point
(yr 1) which determines 0[2(«)*
Statement (7) is accordingly proved.
We can now prove the lemma.
Let the constants a»- be chosen so near unity that on A[2(a) J(n) assumes a
proper absolute maximum at the point (717) which determines g{2(a). Neigh¬
boring (7ri) the points on A[2(«) form what has been termed in Theorem 1.2 a
“non-tangential” submanifold Z of 722r+1(a) belonging to It follows
326
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
from Theorem 1.2 that if c is the length of 0[2(a), and if e is a sufficiently small
positive constant, the locus
J(tt) = c — e [onA[2(a)]
s a spannable ( k — l)-cycle uk- 1 belonging to the critical set a determined by
g[ 2 (a). Moreover the cycle uk- 1 is bounding below c, in fact bounds the chain of
points on A [ 2 (a) at which J (71-) is at most c — e.
Thus A [2(a) is a linking &-cycle on n2r+1(a) belonging to the critical set a on
U(a) determined by 0[2(a), and the lemma is proved.
Symmetric chains and cycles
6. The proof of the existence of linking cycles belonging to the geodesics
gTij{a) is most conveniently made with the aid of the symmetry properties of
Em(a). Before turning to the main theorems of this section we recall certain
facts about quadratic forms.
Let Q(v) be a quadratic form in n variables (v). The constants A and sets
(v) 9^ (0) which satisfy the conditions
Qvi - 2A Vi = 0 (t = 1, • • • , m)
are respectively the so-called characteristic roots and characteristic solutions
belonging to Q. If Q has the index k , there will be k mutually orthogonal char¬
acteristic solutions
(»*) (h = 1, • • • , *)
belonging respectively to characteristic roots A* which are negative. The
fc-plane Lk consisting of points (t>) linearly dependent on the sets (vh) will be
termed the index hyperplane belonging to Q. This index hyperplane is uniquely
determined by Q. Moreover on Lk for normalized sets ( vh)y
(6.1) Q(v) = (h = 1, • • • , k; i = 1, • • • , /*)
as is well known. We thus see that Q(v) is negative definite on Lk .
Any orthogonal transformation V of the variables (v) leaves the characteristic
roots of Q invariant. If V leaves Q invariant as well, it transforms characteris¬
tic solutions of Q into such solutions. We embody these facts in the following
lemma.
Lemma 6.1. An orthogonal transformation of the variables (v) which leaves the
form Q(v) invariant leaves the corresponding index hyperplane Lk invariant. On
Lk, Q(v) is negative definite.
We turn to a function /(t>) analytic in the preceding variables (t>) at (v) = (0),
possessing in (v) = (0) a critical point of index k. We suppose that 0 < k <
andthat/(0) = 0. We set
(6.2) Q(v) — fvivj(0)viVj
C i , j = • ' • > /*)
[6]
SYMMETRIC CHAINS AND CYCLES
327
and let Lk be the index hyperplane corresponding to Q. We term Lk the index
hyperplane belonging to/ and the point (v) = (0).
Suppose that (t>) = (0) is a non-degenerate critical point of /. We shall
investigate the sign of / on normals to Lk near the origin. To that end let an
orthogonal transformation from the variables ( v ) to the variables ( x ) be used, of
such a nature that Q{v) takes the form
Q(v) = \tx] (t = 1, • • • , p).
We suppose moreover that Lk has been carried into the &-plane of the first k
axes in the space (x). The roots Xi, • • • , \k are accordingly negative, and the
roots \k+i, • * • , positive (0 < k < p).
Let ah • • ■ , ak and ck+ 1, * • * , be parameters such that
(6.3) Xja i + * * • + X*.a* = — 1, XA.+1c^+1 -j~ * • * + XMcJ = 1.
Set
(6.4)
(7fli
(i = 1, • •
• , k),
pc,
+
II
■ , m)-
If the parameters cr, and Cj in (6.4) are held fast while p is varied, (6.4) defines a
straight line normal to Lk at the point at which p = 0 in (6.4). Moreover any
normal to Lk can be expressed in the form (6.4). On such a normal / takes the
form
/ = P 2 - + F(p, cr),
where F(p, a) is a power series in p and a involving terms of at least the third
degree, with coefficients which are polynomials in the parameters and Cj}
subject to (6.3).
The equation
0 = / = p2 - cr2 + F(p, a)
possesses solutions of the form
P = -<?- + ‘ ,
where the terms omitted are power series in a of degree higher than the first
with coefficients which are analytic functions of the parameters a* and satis¬
fying (6.3). From (6.5) we see that on each normal to Lk sufficiently near the
origin but not through the origin, / vanishes just twice. On each such normal
the segment on which/ < 0 includes the point p = 0 on Lk. On the normals to
Lk through the origin, /is never negative sufficiently near the origin.
These results lead to the following lemma.
328
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
Lemma 6.2. Letf(v i, • • • , v^) be a f unction which is analytic in the variables (v)
when (i>) = (0), and for which ( v ) — (0) is a non-degenerate critical point of index k,
with 0 <k < y. Suppose moreover that f (0) = 0.
The domain f < 0 neighboring (v) — (0) can then be continuously deformed among
points at which f < 0 onto the index hyperplane Lk belonging to f and (v) = (0), by
moving an arbitrary point P at which / < 0 along the normal to Lk through P to the
foot of the normal on Lk, moving P at a rate equal to the distance to be traversed.
The transformations V* . Corresponding to a fixed integer k between 1 and
m + 1 inclusive, let V* denote the transformation
(6.6)
w'i = w% (i = 1, • • • , k - 1, k + 1, * • • , m + 1),
w'k = —wk.
The ra-ellipsoid Em(a) is invariant under each transformation Vk. It will be
convenient to suppose that the polyhedral complex K of §1, Ch. VIII, to which
Em(a) is supposed homeomorphic, has been so divided into cells that its cells are
carried into cells of K under transformations Vk.
By the transformation Vk as applied to a point (tt) we mean the transformation
effected by applying Vk to each of the vertices of (tt).
We modify the division of Kp and TP into cells as follows. Let the zth co¬
ordinate Wi of the (?th vertex of a point (tt) on Kv be denoted by
w\ (q = 1, • • • , p;i = 1, • • • , m).
We begin as in §1 of Ch. VIII, sectioning and subdividing that portion IP of Kv
for which
w\ ^ 0.
We add the hyperplanes w J = 0 to the sectioning hyperplanes. We then apply
the transformations Vk to the resulting cells of Hp , thereby obtaining a cellular
division of the whole of Kp. In making any further subdivision of Kv we first
subdivide Hp , and then apply the transformations Vk to the resulting cells,
thereby obtaining a subdivision of the whole of Kp, With Kp so divided we
define IP as in Ch. VIII.
A chain zp on IP will be called symmetric if an arbitrary t-cell of zp is trans¬
formed into an i- cell of zp under each transformation Vk. An homology will be
termed symmetric , if the cycles involved are symmetric and the chain bounded
can be chosen so as to be symmetric. A deformation will be termed symmetric
if points (t) which are images of one another under a transformation Vk are
replaced at the time t by points (tt) which are likewise images of one another
under Vk. The Veblen- Alexander process of reduction of a fc-cycle wp on IP
to a fc-cycle of cells of IP will lead to a symmetric homology if wv is symmetric,
provided 0-cells of wp which are images one of the other under transformations V
are assigned to 0-cells of IP with the same property.
We shall now prove the following lemma.
[6]
SYMMETRIC CHAINS AND CYCLES
329
Lemma 6.3. Corresponding to a geodesic g = g^ia) and constants ax sufficiently
near unity , there exists a symmetric proper section S of Kr belonging to g. The
integer p can be taken as any multiple of 4 r, and the parameters ( v ) regularly rep¬
resenting S neighboring the point (v) = (0) determining g , can be chosen so that
the transformations Vk correspond to orthogonal linear transformations of the vari¬
ables ( v ).
We shall give the proof of the lemma for the case of the geodesic g — g„ t 1 (a),
and refer the neighborhood of g to coordinates
Or, yh • • • , yn) (n = m - 1)
as in §3. In the space ( x , y), g can be represented by a segment of the x axis on
which
0 < x ^ 2nr.
We take t he q\h manifold of S as the locus on which
(0.7) r = 0/ = 1 , ■■■ ,p)
neighboring g . We see that so defined is a symmetric proper section of Rp
belonging to <7, provided the constants at are sufficiently near unity and p is a
multiple of 4r.
Let the dh coordinate yx of the point on the gth manifold (6.7) be denoted by
(6.8) y\ (q = 1, * • • , p; i = 1, • • • , n).
We denote the ensemble of the variables (6.8) by (v). The variables (v) para¬
meterize S. On S, in terms of the variables (6.8), and for values of k on the range
1, • •• , m — 1, the transformation Vk takes the form
(6.9) ' y'k* = -yl (g = 1, , p).
The remaining transformations, Vm and interchange the manifolds (6.7)
without changing the coordinates (y) of points thereon. The transformations
Vm and Ym + i when applied to S thus define substitutions of the form
(6.9) " y'y = y\ (i = 1, , a)
in which q and qf are integers independent of i.
The lemma follows from the nature of the transformations (6.9).
The number /i of variables (6.8) is at least 4 r(m — 1), since p is assumed to be a
multiple of 4r. On the other hand the index kri3- of gr{i(a) is at most
K,m+1 = (2 r + 1 )(m - 1) < 4 r(m - 1).
Hence
0 < k^ < p,
a fact of value in the application of Lemma 6.2.
330
SOLUTION OF THE POINCARft CONTINUATION PROBLEM [ IX ]
We continue with the following lemma.
Lemma 6.4. Suppose the constants Oi > * • • > am+i are so near unity that the
geodesic g — g^fa) is non-degenerate and possesses a symmetric proper section S on
Rp. Let c be the length of g} k its index , and a the critical set determined by g.
(a) . Any symmetric , non-spannable j-cycle zp on IP, below cy sufficiently near
<rp, for which j ^ k — 1 , will possess a 2-fold partition symmetrically homologous
to zero , below c, on Tl2p neighboring <r2p.
(b) . Any symmetric j-cycle zp on IF sufficiently near apy for which j > 0, will
possess a 2-fold partition symmetrically homologous to zero on n2p neighboring a2p.
We shall first prove (a). Our previous analysis would make it a relatively
simple matter to show that (a) holds, were the condition of symmetry removed.
Our problem is then to review the homologies by virtue of which we know that
(a) holds disregarding symmetry, and then to show that the homologies involved
may be taken as symmetric.
It will be sufficient to prove the lemma for the case that zp is a /-normal cycle.
For in any case an application of the deformation 0p(t) of §5, Ch. VIII, would
deform the given cycle, below c, neighboring crp into a ./-normal cycle. More¬
over an examination of the definition of the deformation Bp(t) shows that this
deformation is symmetric.
Suppose then that zv in (a) is /-normal. By virtue of (11.4) in Ch. VIII we
have an homology
(6.10) zp — ' bp(zp ) + twp (on Np , below c)
where bp(zp ) is the extremal projection of zp on S, and wp is a (k — l)-cycle on zp.
The analysis of the homology (11.4) in Ch. VIII shows that this homology is
symmetric if zv and S are symmetric, and that wp is symmetric. That the 2-fold
partition of Zwp is symmetrically homologous to zero on n2p, below cf neighboring
cr2p, follows from the symmetry of the deformation used in Theorem 7.1 of Ch.
VIII to prove the 2-fold partition of £wp homologous to zero.
The lemma will follow from (6.10) after we have proved that
(6.11) bp(zp) ~ 0 (on S, below c)
neighboring c, and that this homology is symmetric.
To that end let ( v ) be a set of parameters representing S as in Lemma 6.3, and
let f(v) be the value of /( tt) at the point (7 r) determined by ( v ). Let Lk be the
index hyperplane belonging to/ and to the critical point (i>) = (0). It follows
from Lemma 6.3 that Lk is symmetric. It follows from Lemma 6.2 that the
cycle bp(zv) can be symmetrically deformed along the normals to Lk, below c, into
a cycle uv on Lk.
On Ljt, f{v) assumes a proper, non-degenerate maximum at the origin. Let
ip{a) represent the value of f(v) on Lk in terms of parameters (a) regularly
representing Lk. If e is a sufficiently small positive constant, the locus
p — c, — e
16]
SYMMETRIC CHAINS AND CYCLES
331
on Lk neighboring the point (v) = (0) will be a spannable ( k — l)-cycle belonging
to g, as follows from Theorem 7.5 of Ch. VI. If up is sufficiently near the point
( v ) = (0) on Lh> and the constant e is sufficiently small, up can be symmetrically
deformed on Lk into a ./-cycle v p on <p = c — e, below c, with the aid of trajectories
on Lk orthogonal to the manifolds <p constant. We now give the locus <p = c — e
a symmetric division into cells, and infer from the Veblen- Alexander process
that vp is symmetrically homologous on <p = c — e to a cycle xp of cells of <p =
c — e. But since vp and <p — c — e have dimensionalities j and k — 1 respec¬
tively, and j ^ k — 1 , the cycle xv will either be null, mod 2, or identical with the
cycle = c — e. Since <p = c — e defines a spannable {k — l)-cycle belonging
to <7, and xp is not such a cycle in accordance with our hypothesis in (a), we infer
that xv must l>e null.
Thus (6.1 1 ) holds as stated, and (a) follows from (6.10).
To prove (b) we again make use of (6.10) omitting the condition, below c,
again noting that (6.10) is a symmetric homology, and that possesses a
2-fold partition symmetrically homologous to zero on IP^ neighboring <r2p. We
shall conclude the proof of (b) by showing that
(6.12) bp(zp) — 0 (on IP)
neighboring a7', and that this homology is symmetric.
To that end we symmetrically deform the cycle bp(zp) into a j-cycle uv on Lk
as before. We then use the trajectories orthogonal to the manifolds <p constant
to deform uv symmetrically on Lk into the point ( v ) = (0). The homology
(6.12) thus holds as stated, and (b) is proved.
The proof of the lemma is now complete.
The following hypothesis will be made in several of the following theorems.
It will be validated in §7.
Inductive Hypothesis. If corresponding to an arbitrary positive integer s,
constants ai, • • • , aw + i arc chosen sufficiently near unity) each geodesic gTij{a) for
which r < s will possess a linking cycle on 12.
We let
crij(a)
denote t he length of g and state the following theorem.
Theorem 6.1. If corresponding to the positive integer s , constants ah * • • , am+i
are chosen sufficiently near unity , and each geodesic g J;-(a) for which r < s possesses a
linking cycle on 12, any symmetric j-cycle wp of dimension
j ^ kU - 1,
on which J(ir) is at most a constant ikf, and which is 12- homologous to zero below
rj2(a), will possess a partition which is symmetrically homologous to zero below M.
332
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX 1
We begin by replacing wp by a /-fold partition wq where
t = 4(s - 1)!,
and
g = tp.
This is done in order that q may be a multiple of the integer 4 r corresponding to
each geodesic g\2 for which r < s. We then take the constants ai > • • • > am+ 1
so near unity that the geodesics g\2(a) for which r < s are non-degenerate, possess
the respective indices kri]} have lengths less thancj2(a)> admit symmetric proper
sections S of Rq of the type of Lemma 6.3, and in accordance with Theorem 4.1
are the only closed geodesics with lengths less than cj2(a).
Of the geodesics whose lengths are less than c\ 2 (a), let G\ be the set of geodesics
of greatest length C\. Let aq be the critical set on IF(a) determined by GY
We shall prove the following statement.
(a) The 4-fold partition of the symmetric cycle wq is symmetrically homologous
below M to a cycle w 4 Q below Ci.
Let L be the domain of points (w) on IP(u) below c j. Let Nq be an arbitrarily
small neighborhood of the critical set a 7 determined by GY We make use of
the deformation D * of §4, Ch. VIII, and then of the deformation A q(t) of §5,
Ch. VIII, to deform wq into a cycle zq on Nq + Lqy such that the points of zq
which lie on Nq are /-normal. An examination of the definitions of Z>* and
A q(t) shows that these deformations are symmetric.
We symmetrically subdivide zQ so finely that it can be written in the form
(6.13) z*=u* + v9,
where uq is a symmetric /-normal chain on Ar<7, and vq is a symmetric chain below
Ci. Let xq be the common boundary of uq and vq. The cycle xq will be a
symmetric /-normal (j — l)-cycle on N 9 below cx. It will not be a spannable
cycle belonging to aq, because z q would then be a linking /-cycle belonging to
<j q } and could not be 12-homologous to zero below cj 2 (a)*
If the neighborhood Nq is sufficiently small, the statements in the following
paragraph are true.
Corresponding to each distinct critical set <?% determined by GY the neighbor¬
hood Arq will contain a neighborhood of <jI at a positive distance from the residue
of Nq. The subcycle of x q in none of these separate neighborhoods is spannable,
for otherwise xq would be spannable. It follows from Lemma 6.4 (a), that the
2-fold partition of each of these subcycles bounds a symmetric /-chain on n?9(a)
neighboring a2q below cj. Hence (6.13) can be replaced by a congruence
£2, + y,
in which a2q is the 2-fold partition of zq, j32q is a symmetric /-cycle on II2 q neigh¬
boring a2qj and y2q is a symmetric /-cycle on II2* below cx. It follows from
Lemma 6.4 (b) that the 2-fold partition of a2q is symmetrically homologous to
[7]
THE LINKING CYCLES X rtJ(a)
333
zero neighboring aAq. We conclude that the 4-fold partition of w q is symmetri¬
cally homologous below M to the 2-fold partition of y2q.
Statement (a) is thereby proved.
We now repeat the reasoning used in the proof of (a), replacing c*12(a) by the
length ci of the geodesics of the set Gh and wq by the cycle w4q of (a). By virtue
of the hypothesis that the geodesics g^fa) for which r < s each possess linking
cycles, we see that there are no cycles below c i 12-homologous to zero below
c]2(d) which are not 12-homologous to zero below c\. The cycle w4q must in
particular be f2-homologous to zero below ch since it is ^-homologous to zero
below c 1 2 (a). Let C2 be the set of geodesics g Jy(a) whose lengths equal the maxi¬
mum c2 of the lengths glfa,) less than Proceeding as in the proof of (a) we
can show that the 4- fold partition of wiq is symmetrically homologous below
Ci to a cycle below c>.
Continuing this process we find that, a suitable partition of wq is symmetrically
homologous below c[ 2 («) to a cycle wq' on which J is less than an arbitrarily small
positive constant. The Veblen-Alexander process will then suffice to show that
wq' is symmetrically homologous to a cycle of contracted cells on II9', and thus
symmetrically homologous to zero.
The proof of the theorem is complete.
Note. Let v be the number of geodesics g rl}(a) for which r < s. Let // be the
integer
M - 4>'+10 - 1)1.
The preceding proof shows that a //-fold partition of wv will be symmetricallv
homologous to zero below c\2(a).
The linking cycles \ri:(a)
7. In this section we shall establish the following. Corresponding to a fixed
positive integer s, constants
cl i ^ elm -f i
can be chosen so near unity, that the geodesics gT{i(a) for which r ^ s, possess
linking cycles X L(a) . We begin with the following lemma.
Lemma 7.1. The index of the geodesic <7 f 2 (1 ) is (2s — 1 )(m — 1) — k[2.
Let S be a proper section of R4s(a) belonging to g { 2 (a), set up as in Lemma 6.3.
The parameters (v) representing S are explicitly given by (6.8). The value of
J(tt) at the point (x) determined by (v) will be a function f(v, a) which is analytic
neighboring (v) = (0) and (a) = (1). The form
(7.1) Q(v , a) = fPiVj( 0, a)ViVj
will be an index form corresponding to the geodesic g[2(a)- If the constants
di > * • > am+ 1 are sufficiently near unity, we have seen that the index k*i2 of
334 SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
<7J2 (a) is (2s — 1 )(m — 1). Upon letting the variables tend to unity, we see
that the index k of g[ 2 (1) must satisfy the condition
(7.2) k g (2s - 1 )(m - 1).
If on the other hand we set the last n parameters (v) equal to zero, Q(v, 1)
reduces t o the index form in a fixed end point problem corresponding to a segment
X of gj2(l) length 2x8. Inasmuch as there are (2s — 1 )(m — 1) conjugate
points of either end point of X between the end points of X we must have
(7.3) k ^ (2s - 1 ) (m - 1).
The lemma follows from (7.2) and (7.3).
For constants ax > • • > i sufficiently near unity we shall now define a
symmetric chain UJ2(a) whose boundary yj2 (a) is a symmetric spannable cycle
belonging to g*l2(a).
The chain T8l2(a) and cycle y [ 2 (u) . Set
(7.4) t = 4'+2(*!).
The reason for this choice of i will appear in the proof of Lemma 7.3. Let S
be a symmetric proper section of R*( 1) belonging to 0j2(I), set up as in Lemma
6.3 with parameters (v). Let f(v) be the function defined by J(x) on S, and Lk
the index hyperplane corresponding to f(v) and the critical point (v) = (0). Let
(u) be a set of k parameters regularly representing Lk neighboring the point, say
(u) = (0), which determines <?J2(1), and let <p(u) be the function defined by
J(t) on Lk .
The function <p(u) assumes a non-degenerate maximum 2 xs when (u) = (0).
Moreover Lk and the function <p are symmetric. If e is a sufficiently small
positive constant, the domain
(7.5) ip ^ 2xs - e,
on Lk neighboring ( u ) — (0), will be free from critical points of y ?, save the point
(u) = (0). We divide the domain (7.5) into cells in symmetric fashion, and
denote the resulting fc-chain by PJ2(1), and its boundary by t!2(1). On t J 2 (1)
we have
(7.6) (p ~ 2xs — e.
We record the fact that
r;2(i) -7?2(1) lonn^i)].
For constants a t sufficiently near unity the chain T J 2 (1 ) and its boundary 7 1 2 (1)
will possess central projections on n*(a). We denote these projections by
rj2(a) and 7?2(a) respectively. We have
(7.7) ria(a)-7:t(a)
We shall establish the following lemma.
[on n‘(a)].
[7]
THE UNKING CYCLES X , y(o)
335
Lemma 7.2. For constants a\ > ■ - • > aw + J sufficiently near unity and a suffi¬
ciently small positive constant rj, the locus
(7.7 )' J(n) = e\t(a) - v
on r;2(o) is a spannable cycle belonging to g*12(a), homologous on below c*12(a),
toy It (a).
According to Theorem 3.3, the index k\2 of g{ 2 (a), for constants a i > • • • >
am + 1 sufficiently near unity, is given by the formula
k\2 = (2s - 1 )(m - 1).
By virtue of Lemma 7.1 this is also the index of g9i2( 1), and hence equals the
dimension k of the index hvperplane Lk on which the preceding function <p(u) was
defined.
Now let the section 8 of Rl( 1 ) used in defining Y J 2 ( 1 ) be projected centrally onto
R‘(a). The resulting manifold 8(a) will be a proper section of R*(a) belonging
to g[2(a) if the constants ax are sufficiently near unity. The index hyperplanc
Lk belonging to the function defined by J(tt) on Ar(l) will project centrally into a
regular ^-manifold Lk(a) on 8(a), on which J(tt) will assume a proper, non-
degenerate maximum at the point (w) which determines <7j2(<z), provided the
constants «!>•••> am + 1 are sufficiently near unity. For such constants,
and for a sufficiently small positive4 constant 77, the cycle defined by (7.7)' on
rj2(tt) will be a, spannable (k — 11-cycle belonging to the function defined by
J(ir) on S(a) and to the critical point of this function determined by gsl2 (a)-
That the cycle (7.7)' will be a spannable (k — l)-cycle on U belonging to
g\2(a) follows now from Theorem 11.3 of Ch. VIII.
If the constants ax are sufficiently near unity, one can use the trajectories
orthogonal to the loci, J(w) constant on 1T'2(«), to deform the cycle (7.7)' on
r J 2 (a) below c J 2 («) into 7 J 2 (a) .
The proof of the lemma is complete.
We continue with the following lemma.
Lemma 7.3. If corresponding to a positive integer s} constants a\ > • • ■ >
am+ 1 arc chosen sufficiently near unity , and if the geodesics g ri}(a) for which r < s then
possess linking cycles , the cycle 7 J2(a) will be symmetrically homologous to zero on
the domain J(tt) tk M, where M is the maximum of J(tt) ony\2(a).
Let p = 4s. The integer t of (7.4) and p satisfy the relation
t = up
where
u = 4^{(s - 1)!.
The cycle 7 J2 (a) has been defined with a K -ordering of its points (tt), that is, an
ordering with a definite first, second, • • • , and fth vertex. There will accord-
336
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX J
ingly exist a /x-fold join 712(a) of 7 J 2 (a) on II4a(a) in the sense of §8, Ch. VIII, at
least if the constants a* are so near unity that a succession of p elementary
extremals determined by an arbitrary point (w) of 7 J 2 (a) has a J-length at most p.
The cycle 712(a) is obtained by preferring the </th vertices of the points (w) of
7 1 2 (a) for which q is a multiple of p. Let 7* 2 (a) be the /x-fold partition of 712(a).
The cycle y*l2 (a) will lie on the domain II* (a) on which 7 J 2 (a) lies, and can be sym¬
metrically deformed into 7 { 2 (a) by using the deformation 7? of §7, Ch. Vlli,
holding the common vertices fast. We will thus have the symmetric homology
(7.8) ' y*2(a) ~7l*(a)
on the domain J(t) ^ M of II'(a). Since 7* 2 (a) is a partition of 712(a), we also
have
(7.8) " 7?* (a) * 712(a).
For constants a 1 > • • > am 1 1 sufficiently near unity g\ 2 (a) possesses a linking
cycle, as we have seen in §5. Hence any spannable cycle belonging to 0j2(a),
or cycle 12-homologous to such a cycle below c[2 (a), will be S2-homologous to zero
below cl 2 (a). But 7 J 2 (a) is such a cycle, according to Lemma 7.2, and 712(a) is
another such cycle by virtue of (7.8).
We can apply Theorem 6.1 and the appended note to 712(a), and infer that
its g-fold partition y?2 (a) is symmetrically homologous to zero on the domain
J(tt) ^ M . According to (7.8)' the cycle y{ 2 (a) must also be symmetrically ho¬
mologous to zero on the domain J(tt) g M , and the lemma is proved.
We are led to the following theorem.
Theorem 7.1. If corresponding to a prescribed positive integer s, the geodesics
grij(a ) possess linking cycles when r < sf and if the constants ax > • • • > am+i are
sufficiently near unity, the cycle 7 J 2 ( 1 ) will be symmetrically homologous to zero on
the domain ./( tt) S 2ws — ef where 2tts — c is the value of J( w) on 7 J 2 ( 1) .
It follows from the preceding lemma that there exists a symmetric chain such
that
w(a) — > 712 (a) [on I T(a)]
where w{a) is a chain on which J(t) is at most the maximum M of J(j) on 7 { 2 (a),
provided the constants ax > • • • > am+ 1 are sufficiently near unity. But if the
constants a* are sufficiently near unity M will be so near 2tts — e that the central
projection of w(a) on IT(1) will be a chain z on which J(tt ) < 2ws. We thus
have
2 -* 7 * 2 (1 ) [on IT(1), below 2ir*].
We now use the symmetric deformation D* of §4, Ch. VIII, with p — t to
deform z on II*(I) into a chain u below 2vs — e. The cycleyj 2(1) will thereby
generate a deformation chain v, and we have
u -(-
where u + v is a symmetric chain on which J{ t) g 2ns — e.
[on II'(l)]
[7]
THE LINKING CYCLES X7, , (a)
337
The proof of the theorem is now complete.
The linking cycles XJ2(a). Under the hypotheses of the preceding theorem
there exists a symmetric chain on lF(l) on which J(tt) ^ 27rs — c, and
which is such that
(7.9) ' M;2(l) * y 1 2 ( 1 ) •
For constants sufficiently near unity the central projection of 2 (1) on lF(a)
is well defined, and will be denoted by M J 2 (a). Moreover on M J 2 (a),
JM < cl.2 (a)
if the constants are sufficiently near unity. We will then have
(7.9) " Mt2(a)—y912(a) [below c\2 (a)].
On the other hand we have seen in (7.7) that
(7.10) T*12(a) -7 i2(o)
where TJ2(a) is a symmetric chain on which J(w) assumes a proper absolute
maximum c[ 2 («) at the point (tt) determined by (a) ; and the locus
,/(tt) = cl 2 (a) —7] (7) > 0)
on i'i 2 (a) is a spannable cycle belonging to (a) if the constant 77 is sufficiently
small, and the constants «i > • • • > aw f 1 are sufficiently near unity.
We set
(7.11) x;2(a) = MiAa) + r;2(ttj.
Except in the case s = 1, the definition of X J 2 (a) has been made to depend upon
the inductive hypothesis that the geodesics gri}(a ) for which r is less than a
prescribed integer 5 possess linking cycles if the constants ax > * • • > am+ 1
are sufficiently near unity. For constants go > • • > am+1 sufficiently near
unity the cycle Xi2(u) is then a symmetric linking cycle belonging to g 1 2 (a).
In order to define the cycles X[;(a) in general, we introduce a deformation
Rpq of points on Em(a).
The deformation Rpq. We begin by defining a deformation of the space (w)
in the form of a rotation. In this deformation the time t shall increase from 0 to
I inclusive. A point whose coordinates (w) afford a set (z) when t — 0, shall be
replaced at the time t by a point (w) such that
wp — zp cos 7 rt — zq sin irt (p 5* q),
(7.12) wq = zp sin 7 rt + zq cos tt t (0 g t 1),
Wi = Zi,
where p and q are two distinct integers on the range 1, • • • , m -f- 1, and i takes
on all integral values from 1 to m + 1, excluding p and q.
The deformation Rpq of Em(a) is now defined as a deformation in which each
338
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX J
point (w) on Em(a) moves so that its central projection on Em{ 1) is subjected to
the deformation (7.12). Under Rpq
(7.13) ' Wp = ~Zg, Wg = Zpf
when t — 1/2, while when t = 1
(7.13) Wp = Zpj Wq = Zq.
By the deformation Rpq of points (tt) on IT (a) we mean a deformation in
which the vertices of (w) are deformed on Em(a ) under Rpq. A chain on IT (a)
whose central projection on IT(1) consists of inner points of II* ( 1 ) will thereby
be deformed under Rpq so as to remain on I I* (a), provided the constants <7t are
sufficiently near unity.
If w 8 is a symmetric A>chain on lT(a), the initial and final images of we under
Rpq are identical. We let
(7.14) RpqwB
denote the deformation ( k -f l)-chain derived from w 8 under R,pg. If w* is a
&-cycle, the chain (7.14) reduces to a (k + l)-cycle, mod 2.
The cycles X8 For positive integers y < v ^ m + 1 we now set
(7.15) A8,(a) = Rp, p — i * • ■ RnR2iRv,y-i * * * R*sRz2^l 2 (&)•
If y = 1, the first symbol on the right is RVi „_ i. The successive operations of
forming deformation cycles in (7.15) are to be performed in the order jF?82, R43
etc., each operation producing a cycle of one higher dimension.
Referring to (7.11) we set
(7.16) ' Ml. {a) = Rp.p-r ••• U,,-i ••• R32M J 2 (a)
and
(7.16) " r;r(o) = Rp'p-i ••• RtiR,,,- 1 ••• ^32r;2(a).
We see that
(7.16)'" KAa) = + r;,(o).
In each of the preceding deformations the time t runs from 0 to 1 inclusive.
Denote the point into which a point (t) is deformed at the time t under Rpg by
Kf r.
Let 7tJ2 (a) be the point on A J2 (a) which determines gl2(a)- We now set
(a) = RV.l-i • • • *;,(«)•
We observe that the point 7r8 „(a) lies on A8 ,(a), and determines £8 ,(a).
We shall prove a lemma concerning r8 „(a). In this connection we point out
that the definition of r8 „(a) is independent of any inductive hypothesis.
[7]
THE LINKING CYCLES X^.(a)
339
L£mma 7.4. On each chain T * „ (a) for which s is a prescribed positive integer
J (7r) will assume a proper , absolute maximum equal to the length c * „ (a) of g* „ (a), at
the point 7r* „ (a), provided the constants aY> • * * > Om+i are sufficiently near unity .
In proving this lemma we shall make use of the following property of ellipses.
Corresponding to any positive constant a, there exists a positive constant e
with the following property. Let Ef and E" he two ellipses in the xy plane with
centers at the origin but with arbitrary orientations. Suppose no points of En
are exterior to E' . Let b' and 6" be segments of Ef and E"> respectively, which
subtend a common angle at the origin, in magnitude at least a. If the ellipses
E' and E" possess semi-axes which differ from unity by at most e, the lengths
0' and of the segments b' and b" respectively satisfy the condition
(7.17) ^
The proof of these statements can be given by elementary methods, and will
be left to the reader.
To establish the lemma we shall make use of a function II (tt) defined as follows.
Let b be any elementary extremal on Em(a) of positive length and with end points
P and Q . Let X be the 2-plane determined by P, Q and the origin. The 2-plane
X will intersect Em(a) in an ellipse. Of the arcs of this ellipse bounded by P and
Q, let b' be the shorter. We term b' the elliptical arc corresponding to b. To
define H(w) we replace each non-null elementary arc of g{ir) by the correspond¬
ing elliptical arc and leave null arcs unchanged. We denote the value of the arc
length J taken along the resulting curve by II (ir). We observe that
JM ^ H(t).
We shall now establish Lemma 7.4 with II (t) replacing ./( tt).
Observe first that //( t) =h J(tt) when (a) == (1). Hence //( tt) has a non¬
degenerate, absolute maximum on rj2(l) when (tt) — 7r[2(l)- Hence for constants
at such that | — 1 | < rjf where 77 is a sufficiently small positive constant, the
function H(t) will have a non-degenerate, absolute maximum onrj2(a) when
(tt) = 7r 1 2 (a) . There is no limitation in this statement on the relative sizes of the
constants a».
Let the constants (a) be chosen so that
ai > ■ • • > am+ 1
and
(7.18) | at - 1 | < y.
For these constants (ah a2, - * • , am+J) we consider the chain
(7.19) F 1 i>{a^ a r, } ayf ay+xf • • • , Qm+i).
The chain (7.19) can be obtained from the chain F[ 2 with the same arguments, by
using the formula
r;v = Rt >-x P32r;2.
340
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
But for the arguments in (7.19) the deformations here involved become rotations
of the vertices of points (tt). For these arguments H( ir) assumes a proper,
absolute maximum on rj2 equal to the length of an ellipse with semi-axes l/aM
and 1/a,. This length equals the number
CO7'- «*•••» Om+i) = c’.(a),
and is the length of
(7.20) Q 12(^mj y dyy + y + 1)*
We see then that II (tt) assumes a proper, absolute maximum relative to its
values on (7.19) at each point (w) on the cycle
(7.21) l£y,,-l ' * ' R 327r12(flM> av> * * * 9 ap+\> 1 am-fl)’
This maximum equals c * v(a).
We shall next prove the following statement.
(a). For points (t') on the chain
(7.22) I , a-M, ^n+ 1> y ^m-f-i) (m ^ v)
the function H( tt') assumes a proper , absolute maximum c*„(a) at the point (n')
which determines the geodesic
9\ y > &m + 1) j
provided the constants cq > • • • > am+i are sufficiently near unity.
We shall prove (a) for the case s = 1 . The proof for a general s is not essen¬
tially different.
In proving (a) we shall compare each point (tt') on the chain (7.22) with its
central projection (t) on the chain (7.19), taking s = 1. Points (71- ) on (7.19)
will be divided into two classes. The first class shall consist of the points (7 r)
on the cycle (7.21), while the second class shall consist of the remaining points
on (7.19).
Points (t) on (7.21), 5=1. For such points (w), g( ir) is an ellipse obtainable
from the ellipse
(7.23) gi yiany a„ , a> + 1, ' , am+ 1)
by a rotation in which the intersection of the ellipse with the wx axis is fixed. For
such points (7r), H( it) equals the length of this ellipse, namely
(7.24) C(ai, • • • , am+i) = cj„(a).
Let 7(71-) be the central projection of the preceding ellipse g(ir) on the m-
ellipsoid
(7.25)
Em(&ny y Qpy &n + ly * * * >
[7]
THE LINKING CYCLES Xj .(a)
341
Let (7r') be the central projection of (tt) on the chain (7.22). The value of
//( 7r') is the length of 7(7r). The ellipse g{i r) lies on the ra-ellipsoid
(7.26) &VJ i +
Its center is at the origin, it intersects the t/;i axis, and it lies in the space of the
wif • • • , axes. Now p < v and aM > a*. We see that 7(71-) and g(ir) have
their intersections with the w\ axis in common, but that 7(71-) is otherwise interior
to *7(71-), except in the special case where g( t) and 7(?r) are the ellipse (7.23).
But the length of the ellipse g(z) is c\v(a) and the length of the ellipse 7(7r) is
//( 7r'). Hence
(7.27) H( tt') < cj.(a)
for points {*’) which project centrally into points (tt) on (7.21), except in the
case where (7 r') determines the ellipse (7.23).
Points (7 r) on (7.19) but not on (7.21). For such points (t),
(7.28) //(tt) < c;„(a)
as stated in connection with (7.21). But if each elliptical arc P' determined by
such points (7 r) be compared with its central projection p" on the m-ellipsoid
(7.25), it follows from (7.17) that if 77 in (7.18) is sufficiently small
(7.29) tf(ir') ^ //(tt).
From (7.28) and (7.29) we see that
(7.30) H(tt') < c;v(a)
for points (7 r') on the chain (7.22) which do not project centrally into points (7r)
on (7.21).
Statement (a) follows from (7.27) and (7.30).
We shall now prove statement (P).
(P), The function H{ tt) assumes a proper, absolute maximum c* „ (a) on the chain
(7.31) r;,(ai, • • * , a.+i) = r;,(a) 0* < *)
at the point t * , (a) thereon which determines g^ „ (a) .
We shall compare the chain (7.31) with the chain
(7.32) IV (aM, > > u*n-fi).
The latter chain is given by the formula
1% v ~ Rn. n~] ^21^1 v
where the arguments in T{ v are the same as in (7.32). It follows from (a) that
H(w) assumes a proper, absolute maximum c*„(a) on the chain (7.32) at each
point of the cycle
(7.33) /?M, • • • R2 l^rj * • • >
342
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
For 5=1 the curves g(ir) determined by points (t) on (7.33) consist of ellipses
obtainable from the ellipse
9l y (af*t t &M + lf i 0»»+l)
by a rotation in which the points of intersection of this ellipse with the wv axis
are fixed. The central projections of these ellipses on the ellipsoid Em(a) will be
ellipses of lesser length, except for the ellipse
9 ft yfauf ) tiftt &n + h t 1)*
The last ellipse is identical with the ellipse g\ „(a), and has the length cl, (a).
For points (7 r) on (7.32) which are not on (7.33),
(7.34) H( tt) < C;v(a)
as follows from («). But if (w) is any point on the chain (7.32), and (tt') its
central projection on (7.31), we have
//(*■') ^ //(tt)
in accordance with (7.17), provided the constant 7? in (7.18) is sufficiently small.
Hence for points (irf) on (7.31) whose central projections (tt) do not lie on (7.33),
we have
H{ TT') < cl, (a).
Statement (0) follows from this result and the result of the preceding para-
graph.
To return to the lemma we observe that
JM = H( tt) = <,(a)
at the point 7r*„(a) on r*„(a) which determines £*,(a). By virtue of state¬
ment (/8),
(7.35) H( tt) < cl, (a)
at all other points (71-) on T* „(a) if the constants ax> • • • > am+iare sufficiently
near unity. But J(t) g H(w), so that (7.35) gives us the relation
JM < cl, (a) [(ir) ^
for (?r) on r^,(o).
The proof of the lemma is now complete.
We continue with the following lemma.
Lemma 7.5. Points (w) on the chain r* ,(1) neighboring 7r* „(1) make up a non -
tangential manifold 2, belonging to g 1,(1) in the sense of §1. The dimension of 2
equals k * , .
The lemma is true of r[2(l) by virtue of the definition of rj2(l) as the
domain (7.5).
[7]
THE LINKING CYCLES \ri}(a)
343
Let (v) be a set of k[2 parameters regularly representing f;2(1) neighboring
7T 1 2 (1)> with (u) = (0) corresponding to ttJ2(1) Recall that
(7.36) r;,(l) = R" R2lRv, R32T{2( 1).
Let the time t in the respective deformations employed in (7.36), taken in the
order written, be denoted by
(TM~ 1> ' ’ ’ > Tl> U-ly * * * , tl).
Recall that 0 ^ ^ 1 for each such parameter. The general point (t) on r®„(l)
is obtained from an arbitrary point ( v ) on r J 2 (1 ) by subjecting that point to
R32 up to a time tlf subjecting the resulting point to R4 3 up to a time t2f and so
on until all the deformations in (7.36) have been employed, the final deformation
R continuing up to a time r„_ 1. The general point (w) is thus representable
by means of the parameters
W = (ru • * * ,
(0 = (L, * • ; C-2),
and the parameters (v) of the initial point. In particular the point tt*„(1) on
r;„(l) is determined as above by parameters (r) and (t) each of which equals
1 /2, and parameters (v) = (0). We shall prove the following.
(a). In terms of the parameters (t), (t), and (v)y r*„(l) is regular at the point
7r * „ ( 1 ) on F * ,, ( 1 ) which corresponds to the parameter values
(7.37) (v) - (0), (0 = (*), (r) = (§).
We shall establish (a) by showing that in the space of the points (tt) the
directions tangent to the parametric curves on F* „(1) through the point 7r® „(1)
are independent. Of these directions, those involving the variables (t>) alone
are independent among themselves, since the same is true of rj2(l), and since
the deformations in (7.36) subject the vertices of points (n) to a rigid motion.
We consider the curves on which the parameters (t) and (r) vary. In terms
of the parameters (t), (r) and (v) let Mq be the manifold on the unit ra-sphere
on which the gth vertex of the point (tt) on T^(l) varies for parameter values
( t)y (r), and (v) near the values (7.37). Let Af and A" be the intersections of
the positive w M and wy axes with the unit ra-sphere. Of the manifolds Mq let
L ' and L" be two particular manifolds which pass through A' and A” for pa¬
rameter values (7.37). Let the parametric curves on L" through A " on which
one only of the parameters
t\y * y tv—2y r 1, , T n~~ 1
vary and on which the remaining parameters have the values (7.37), be respec¬
tively denoted by
(7.38) h\} • • • , 2, k\y • • • ,
344
SOLUTION OF THE POINCARft CONTINUATION PROBLEM [ IX ]
On taking account of the deformations used in (7.36) one sees that in the neigh¬
borhood of A " the curves hh ■ • • , consist respectively of segments of the
circles
(7.39) 9\ vi ‘ y 9fi- 1 , vy 9 1 , ^ ’ > 9 v- -l,vt
while the curves kh ■ • • , A^_i reduce to the point A". If
(7.39) become the set
9lv) ’ * * ) 9 V~ 1 ,
To determine the curves (7.39) let Qi denote the intersection of the unit
ra-sphere with the positive Wi axis. Under R3 2, Q2 is rotated into the point
Q3, reaching Q3 when h = 1/2. The path of Q2 is thus a segment of g\ 3 . Under
Rizy Q2 is fixed, while Q3 is rotated into Q4 when t2 — 1/2. Thus the path
is rotated into the path <724 when t2 ~ 1/2. The successive application of the
deformations RM, • • ■ , Rv, *-1 up to times t3 = • * • = ^__2 =1/2 respectively
will rotate gl24 into the path g2l). The deformations RiU • • • , i?Pt are now
to be successively applied to g\ v up to the times n = • • • = rM _i = 1/2 re¬
spectively. Of these deformations Rn alone affects g\ „, rotating g\ „ into g\
Thus the parametric curve hi is a segment of g\v as stated. Similar reasoning
will establish that the remainder of the parametric curves (7.38) lie on the
corresponding circles in (7.39).
Let
v y v
respectively, denote the submanifolds of 2 through the point 7r*„(l) on which
the parameters (t), (r), and (t>) vary neighboring the sets (7.37). We have seen
that the manifold 2V is regular. That the manifold 2, is regular follows from
the mutual orthogonality of the circles (7.39).
To show that 2r is regular at x* „(1) we consider the parametric curves on L',
w hich pass through A ' corresponding to the values (7.37), and on which one only
of the parameters,
Tl, • • • , Tm_ 1,
varies. Following the trajectory of the point Qi under the successive deforma¬
tions in (7.36) one finds that these parametric curves consist respectively of
segments of the mutually orthogonal circles
(7.40) </i„, • • • , „ [(a) = (1)].
That the manifolds 2* and 2r have no tangents in common at the point w* ,(1)
follows from the fact that the curves kh • • * , in (7.38) which result from a
variation of the respective parameters (r) of 2r reduce to points, while the
remaining curves in (7.38) which result from a variation of the respective
parameters ( t ) on 2* have mutually orthogonal directions. Thus the sub¬
manifold 2* of 2 through 7r*,(l), on w-hich (t) and (r) alone vary, is regular in
terms of the parameters (£) and (r).
[7]
345
THE LINKING CYCLES Xj (a)
That the manifolds E* and Ev have no tangent line in common at 7r* „ (1) follows
from the fact that on analytic curves tangent to Ev at (v) — (0), J{w) assumes a
non-degenerate maximum when (v) = (0), while on 2* J(tt) is constant. Thus
the directions of the tangents to the respective parametric curves of E through
* * „(1) are independent .
Statement (a) now follows.
We continue with a proof of statement (0).
(0). The manifold E is a n on -tangential manifold belonging to
Let the manifolds on R on which lie the successive vertices of a point (7 r) on
Ety ETy or E0 be termed vertex manifolds. For points (w) neighboring 7r*„(lj
these vertex manifolds are readily seen to be orthogonal to 0*„(1). It follows
that the corresponding vertex manifolds of E are orthogonal to gr*„(D- Upon
recalling the definition of non-tangential manifolds belonging to gl „(1) one sees
that E must belong in that category, and (0) is proved.
We complete the proof of the lemma by proving statement (7).
(7) . The dimension j of F * „ ( 1 ) equals A* * „ .
First recall that the dimension of rj2(l) is AJ2- The numt>er of parameters
(v) thus equals A:J 2 . The number of parameters (t) and (r) is ^ + v — 3, so that
J — k 1 2 T m T v ~~ d.
But
A* J 2 - (2.s - 1 )(m - lj.
Hence
j — (2.s — 1 )(/a — 1) + M + v —
— m v v — 4 + 2(s — !)(/// — 1 ).
rfhis is the value of k‘l „ as stated.
The preceding lemma leads to the following:
Lemma 7.6. Let s be a prescribed positive integer , and c * v(a) the length of g l „(a).
If the constants a\ > • • • > am+\ are sufficiently near unity and e is a su fficiently
small positive constant the locus ,
J{ir) = c*,(a) ~ e
on F* y(a)y will be a spannable cycle belonging to gl „(a).
We shall prove this lemma with the aid of Theorem 1.2.
To that end first observe that the points (7 r) on F*„(a) neighboring 7 r*„(a)
form an analytic manifold E(a). According to Lemma 7.5 this manifold is a
“non-tangential” manifold, belonging to <7*„(1) when (a) = (1). By virtue of
the definition of such non-tangential manifolds one sees that r* r(a) will remain a
non-tangential manifold belonging to glv{a ), if the constants at are sufficiently
near unity.
346
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
According to Lemma 7.5 the dimension of 2(a) will equal v, and thus equal
the index of glv(a). According to Lemma 7.4, J(ir) will assume a proper,
absolute maximum c * „ (a) on r * „ (a) at the point t * „ (a) if the constants ax >
• • • > am4i are sufficiently near unity.
The lemma follows from Theorem 1.2.
We come to a basic theorem.
Theorem 7.2. Let s be a prescribed positive integer . If the constants ax >
• * * > am+ 1 are sufficiently near unity , the cycles \*y(a) exist and are linking
cycles belonging to the geodesics glv(a ), for all integers r ^ s.
We first consider the case s = 1.
If the constants ax > • • • > am+i are sufficiently near unity, the following
statements are true. The cycle X}2(a) will be a linking cycle belonging to
g\2(o)y the definition of X}2(a) depending upon no inductive hypothesis. If e
is a sufficiently small positive constant, the locus
(7.41) J( tt) = cl „ (a) - e
on T lv(a) will be a spannable cycle belonging to g\ v(a)y according to Lemma 7.6.
On rj„(a), J(ir) will assume a proper, absolute maximum c^(a) at the point
7J-* v(a). The chain Af J2(l) of (7.9)' exists, by virtue of Theorem 7.1, and the
chain Af*„(a) is then defined by (7.16)' for v > 1, and will lie below c* „(a).
The cycle \\ Ja) can now be defined by the congruence
Xj,(«0 = Ml, (a) + Tl,(a)}
as in (7.16)'" We see that the cycle (7.41) on r*,(a) will bound below cl,(a)
on X^v(a). Hence X^„(a) will be a linking cycle belonging to gl,(a). The
theorem is thus true when s — 1,
Proceeding inductively we assume that the theorem is true when ,9 is replaced
by s — 1. This inductive hypothesis enables us to apply Theorem 7.1 and infer
that M\ 2 (a) exists as in (7.9)". The cycle XJ2(a) can then be defined as in
(7.11), and the cycle X£v(a) as in (7.15) We prove that X*„(a) is a linking
cycle belonging to glv(a) as in the preceding paragraph.
The proof of the theorem is complete.
The circular connectivities of the m-sphere
8. Before coming to the problem of the existence of closed geodesics we shall
solve the basic topological problem of the determination of the circular con¬
nectivities of the m-sphere. Recall that these connectivities are the ^-con¬
nectivities of the space 12 determined by an admissible metric on the m-sphere.
An admissible metric is any metric with elementary arcs of the nature defined in
§12, Ch. VIII. These circular connectivities will be independent of the metric
used and, as we have seen, are topological invariants. On Riemannian mani¬
folds the metric can be defined, if one pleases, by the integral of arc length, and
the elementary arcs defined by means of geodesics.
[8]
THE CIRCULAR CONNECTIVITIES OF THE m-SPHERE
347
We shall make use of the notation of the preceding chapter, referring to the
ra-ellipsoid Em(a) and to 12(a). We shall take the constant p which limits the
lengths of elementary extremals as a fixed number such that
< p < -
The circular connectivities of the m-sphere will be found by determining the
^-connectivities of 12(1).
We begin with the following theorem.
Theorem 8.1. Corresponding to any positive integer k , a maximal set of k-cycles,
12 -independent on 12(1), consists of the cycles of the set
M,(l) 0 = 1, 2, • • • ;i,j= 1, • • • , m + 1; i < j)
of dimension k.
Statement (a) will now be proved.
(a). Any k-cycle z on 12(1) is Si-homologous to a linear combination of the k-
cycles of the theorem .
On z suppose that J(w) is less than 27r$, where s is some positive integer.
Without loss of generality we can assume that the elementary arcs determined
by 2 have lengths at most p/2, because in any case a 2-fold partition of z would be
12-homologous to z and have this property.
Let
1 > a\ > • • • > am + ] > 0
be a fixed set of constants, and let
Ul ^ + 1
be a set of constants on the range
(8.0) a< = a< + <(1 - cti) (0g/< 1).
The constants a, will satisfy the condition a,; < 1 so that the m-sphere Etn( 1)
will be interior to Em(a). It follows that the central projection on 12(1) of any
point (7 r) on 12(a) will be admissible; for an elementary geodesic X on Em{a) will
project centrally into a shorter curve p on Em( 1). The elementary geodesic
joining the end points of p on Em( 1) will then be shorter than X. Thus the
central projection on 12(1) of an arbitrary point (w) on 12(a) will be admissible.
Moreover if the above constants a* are chosen sufficiently near unity, the
statements of the following paragraph are true.
The central projection z(a) of z on 12(a) is admissible for constants (a) given by
(8.0). On Em(a) there are no closed geodesics with lengths less than 2ws other
than the geodesics <7 L /<*) for which r < s. The geodesics grii{a) for which r < s
are non-degenerate, and possess the cycles XJ/a) as linking cycles. The value
of J(w) on z(a) is less than 2^5.
348
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
It follows from the theory developed in Ch. VIII that z(a) is O-homologous
to a linear combination, say v(a ), of the cycles X -y(a) for which r < $. There
will then exist a partition, v'(a ) of and a partition z'(a) of z(a), both on a
domain ng(7r), together with a (k + l)-chain w'(a), also on If^ar), such that
(8.1) w'(a) — * v'(a) + z'{a).
If we denote the central projection, on 0(1), of a cycle x of 0(a), by c[x], it follows
from (8.1) that
c[w'(a)] - c[v'(a)\ + c[z'(a)]
so that
(8.1a) c[v'(a)] ~ c[z'(a)] [°n n9(l)j.
Let v(a ) denote the central projection of y(a) on 0(a), and v'(a) the partition of
v(a) on IU(a). If we let t in (8.0) range from 0 to 1 inclusive, the cycle c[v\a)}
will generate a ( k -f- 1) -chain v* on llg(l) such that
v* —* c[v\a)] + t>'(l).
Thus
(8.1b)
1>'(1) ^ c[t/(a)]
[on
n*(DJ.
It follows similarly that
(8.1c)
2'(1) - c[z'(a) ]
[on
n*(i)].
But z'( 1) and t/(l) are respectively partitions of 2(1) and t>(l), so that
(8.1d) z'(i) * 2(1), v'(\) * t>(l).
From the homologies (8.1a) to (8. Id) we see that
(8.2) 2(1) * v(l) [on 0(1)].
But 2(1) is the given cycle 2, and y(l) is a linear combination of the A>cycles of the
set XJ,-(1).
Statement (a) follows then from (8.2).
We continue with a proof of statement (#).
(/9). The k-cycles of the theorem are Q-independent .
Suppose (0) is false, and that a is a proper linear combination of ^-cycles of the
set X U(l), 0-homologous to zero on 0(1). There will then exist a partition of uy
say w, on a domain n<7(l), together with a (A: + l)-chain 2 on IIff(l), such that
2 > w [on ng(l)].
On 2 suppose that J(tt) is less than 2 ts. If q is taken sufficiently large, the
elementary extremals determined by 2 and w will be at most p/2 in length.
[8]
THE CIRCULAR CONNECTIVITIES OF THE m-SPHERE
349
Let w(a) and z(a) be central projections of w and 2 respectively on nv(a). For
constants sufficiently near unity, we will have
z(a) w(a) [on IU(a)].
But if these constants a{ are sufficiently near unity, the ^-cycles of the set XU(a)
for which r ^ s are f2-independent. It is impossible therefore that z(a) — ► w(a).
We conclude that u is not ^-homologous to zero on 12(1), and that (/?) is true.
The proof of the theorem is complete.
The Mh circular connectivity of the //^-sphere is then the number of ^-cycles
in the set X V(l). We thus have the following corollary of the theorem.
Corollary. The Jcth circular connectivity Pk of the m- sphere is the number of
distinct integral solutions i, j, r of the diophanti ne equation
(8.3) k = m + i + j — 4 -f 2 (r — 1 )(m — 1)
in which m + 1 ^ i > j > 0, r > 0, and k and m are fixed .
The sequence of circular connectivities
(8.4) P0P1P2 * * -
can readily be determined from (8.3) for a given m . We give the determination
for the cases m — 2, 3, 4, and 5:
(m = 2) 011212 ;
(m = 3) 00112121212 ;
(m = 4) 0001122212122212
[rn = 5) 000011223221212232212
The numbers underlined represent a group which thereafter repeats periodically.
The first general existence theorem is the following.
Theorem 8.2. Corresponding to any admissible functional J, defined on any
Riemannian manifold R which is the topological image of the m-sphcret there exist
critical sets of closed extremals on R whose kth type number sum is at least the kth
circular connectivity Pk of the m-sphere.
There also exists a number Lk which depends only on R , /, and k , and which is
such that if all closed extremals on R with J~lengths less than Lk are non-degenerate ,
there will be at least Pk non-degenerate , closed extremals on R of index k> with
J-lengths at most Lk.
By virtue of the topological invariance of the circular connectivities there will
exist a set (X)* of Pk cycles on the space SI determined by R and the functional J .
Of the minimum critical values, “determined” in the sense of §6, Ch. VIII, by
linear combinations of cycles (X)*, let Lk be the maximum. It follows from
Theorem 6.5 in Ch. VIII that the minimal set K of closed extremals determined
350
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
by the set (X)* will lie on the domain J S Lk, and will consist of critical sets of
extremals whose fcth type number sum M* is at least P*. In case all closed
extremals with /-lengths at most Lk are non-degenerate, the number Mk is the
number of closed extremals of index k in the set K.
The theorem thus holds as stated.
Topologically related closed extremals
9. Let R ' and R " be Riemannian manifolds of the nature of the preceding
manifold R. Suppose that R' and R" admit a homeomorphism T. Let 12'
and 0" be the respective spaces 12 defined by functionals /' and /" on R' and P".
Identifying Rf and P" and their metrics with the spaces S' and S" respectively of
§12, Ch. VIII, we introduce the conception of points (t) which are admissible
rel r0 as defined in (12.14) of Ch. VIII. Points ( tv ') and (t") on O' and 0"
respectively which are admissible rel r0 will be said to correspond under T if their
vertices taken in some one of their circular orders, direct or inverse, correspond
under T.
To avoid ambiguity a partition on 0' of a cycle z on 12' will be called an
O' -partition. An 12 "-partition of a cycle on 12" is similarly defined. Cycles z'
and z" on 12' and 12", respectively, will be said to correspond after partition if
suitable 12'- and 12 "-partitions of z' and 2", respectively, are admissible rel r0
and correspond under T.
If z is a cycle on 12', two cycles on 12" which correspond to 2 after two 12'-
partitions of 2 are mutually 12 "-homologous. It will be sufficient to prove this
statement for the case where 2 is a cycle on U'p. Let u and v be r- and s-fold
12 '-partitions of 2 which admit correspondents on 12". Let the correspondents of
u and v on 12" also be denoted by u and v. We wish to prove that u * v on 12".
To that end let w be the correspondent on 12" of the 7 s-fold 12 '-partition of 2.
Observe that w is also the correspondent on 12" of the s-fold 12'-partition of u .
By virtue of Lemma 12.2 of Ch. VIII, w is homologous to the s-fold 12"-partition
u of u , so that we have
Hence
Similarly
Hence
w ^ u, u * u
w * u
W * V
U * V
(on 0").
(on Q").
(on 12").
(on 12")
as stated.
As seen in the proof of Theorem 12.1 of Ch. VIII, a set of cycles on 12' which
[9]
TOPOLOGICALLY RELATED CLOSED EXTREMALS
351
satisfy no 12-homology on 12' will correspond after partition to a set of cycles on
12" which satisfy no 12-homology on 12".
Let ( u' ) and (u") be finite sets of 12-independent cycles on 12' and 12" re¬
spectively, with members which correspond respectively after partition. The
sets (u') and (u") will determine minimal sets K' and K” of closed extremals
on W and R" respectively, in the sense of §6, Ch. VIII. We shall then say that
K' and K" are topologically related under the homeomorphism T.
To illustrate this conception we return to the ra -ellipsoid Em(a) and the
space 12(a) determined by the integral of arc length on Em{a). We state the
following theorem.
Theorem 9.1. If the constants ai > • • • > aTn + i are sufficiently near unity , the
minimal set of closed geodesics on Em(a) determined by the linking cycle A^ (a) is
the ellipse glhk(a).
We choose the constants > • • • > so near unity that the principal
ellipses g\ j (a) are non-degenerate, possess the cycles A*, (a) as 12-independent
linking cycles respectively, and are the only geodesics on Em(a) with lengths
less than Sir. Let A be one of the cycles Ajy(a). Let K be the minimal set of
closed geodesics determined by A. Suppose K includes a geodesic glhk(a). In
such a case A must be the cycle A,** (a) as we shall now prove.
Recall that g\k{a) determines the point Tlk(a) on 12(a). By virtue of the
definition of a minimal set of closed geodesics belonging to A there will exist,
among the “reduced new cycles” which are 12-homologous to A, at least one,
say n, for which the corresponding reduced critical set a will include the point
irlhk(a). The cycle n will be 12-homologous among points (t) neighboring a
and below Jijlhk) + e2 to a linear combination L of the cycles A ]3(a). By
virtue of the definition of a reduced new cycle L must include the cycle Aj[fc(a).
We have A * L . Since the cycles \ \ ; (a) are ^-independent this is possible only
if
A = A kk(a).
Hence K consists of the single ellipse glhk(a).
The proof of the theorem is now complete.
Let R be a Riemannian manifold homeomorphic with the ellipsoid Em(a) of
the theorem. There exists a well defined minimal set of closed extremals on R
topologically related to each principal ellipse g\k(a) on Em(a). For A \k(a) will
correspond after partition to a well defined cycle u on the space 12 determined by
the integral J on R. The minimal set of closed extremals on R determined by u
will be topologically related to gl k (a) in accordance with our definitions.
We consider the complete set of cycles A l , (a). It is clear that this set of
cycles determines the principal ellipses on Em(a) as a minimal set of closed
geodesics. A set of cycles on 12 which correspond to the cycles A \i(a) after
partition will determine a minimal set G of closed extremals on R. If we
combine this result with Theorem 6.5 of Ch. VIII, we obtain the following.
352
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
Theorem 9.2. Let R be a Riemannian manifold homeomorphic with an m-
ellipsoid Em(a) for which ai > • • • > am+i. If the constants (a) are sufficiently
near unity , there exists a set G of closed extremals which is topologically related on
R to the principal ellipses on Em(a ), and which has a kth type number sum at least
as great as the number of principal ellipses on Em(a) of index k.
We also note the following. On the space 0(0) determined by the ra-sphere
Em{ 0) the cycles Xj , (0) form a set which is O-independent. The corresponding
minimal set of closed geodesics on Em (0) is the set of great circles on Em( 0) . For
these great circles form a connected set of closed geodesics for which the cor¬
responding critical set on 0(0) includes all points (t) on the cycles X* y(0), while
the cycles X | i (0) are not O-homologous to cycles belowr 2w, since there are no criti¬
cal values below 2t r.
Metric relations between topologically related closed geodesics. Let T represent a
homeomorphism between two Riemannian manifolds R' and R" . Let the
functionals J! and J" be the integrals of arc length on Rf and R " respectively.
If the homeomorphism T can be locally effected by a non-singular analytic trans¬
formation of coordinates, minimal sets of closed geodesics which are topologi¬
cally related on Rf and R" respectively stand in noteworthy metric relations.
Let P' and P" be points wLich correspond on R ' and R " respectively. Let (x)
be local coordinates on R' neighboring P' . By virtue of the transformation T
we can take the coordinates (x) as local coordinates on R" neighboring P".
Let ds' and ds" be the differentials of arc on R' and R* respectively, expressed in
terms of the coordinates ( x ) and their differentials (dx). rThe ratio
ds"
= M(x, dx) [(dx) * (0)1
as
will be a positive continuous function of the variables (x) and (dx) for points ( x )
neighboring P' and sets (dx) ^ (0). It will be homogeneous of order zero in the
variables (dx), and thus depend only upon (z) and the direction T defined at the
point (x) by the differentials (dx). We can regard the preceding ratio as locally
defining a function
% - ■ >
of the point P on Rf and an arbitrary direction F on R' at P.
Let mi a,rid m2 be respectively the absolute minimum and maximum of y(P, F),
for points P on R' and directions T on R' at P. We have
Mi S n(P , F) g M2.
If L' is the length of any regular curve on P', and L" the length of the cor¬
responding curve on R", wre see that
MiL' S' L" ^
We shall pro\e the following lemma.
[9]
TOPOLOGICALLY RELATED CLOSED EXTREMALS
353
Lemma 9.1. Let iY and Q" be the spaces 12 respectively determined by the integrals
of arc length on R' and R". On 12' and 12" let zf and z " be corresponding k-cycles
not U-homologous to zero. If c ' and c" are respectively the minimum critical values
“ determined ” by z' and z", we have the relations
Ml c' ^ c" fl2c'.
By hypothesis the cycles z ' is 12-homologous on 12' to a cycle w ' below cf + e,
where e is an arbitrarily small positive constant. If we take sufficiently high
12 '-partitions of the chains involved in this homology, the resulting chains possess
images on 12" under T, and we see that z” is 12-homologous on 12" to the image w*
on 12" of a partition of w' . But w " will lie below utf' + ^i2e, from which it
follows that
c" ^ M2 c'.
Upon interchanging the roles of Rf and R ", z' and z" , and c' and c" , replacing
ii2 by 1/mi, we see that
c' ^ 1 c".
Ml
The lemma follows from the preceding inequalities.
We state the following theorem.
Theorem 9.3. Let R be a Riemannian manifold which is the non-singular,
analytic homeo morph of the unit m-spherc Eni( 0), and which is such that the ratio of
the differential of arc length on R to the corresponding differential on Em( 0) has an
absolute maximum H2 cmd absolute minimum g i. There exists a set G of closed
geodesics which is u topologically related ” on R to the great circles on Em (0) and which
has the following properties.
(1) . The geodesics of the set G have lengths between 2-Kfn and 2i r/i2 inclusive.
(2) . The kth type number sum of the geodesics of G is at least the number of prin¬
cipal ellipses of index k on any ellipsoid Em{a) for which the constants a i > • • • >
am + i are sufficiently near unity.
(3) . If the closed geodesics on R with lengths between 2ttjh and 2i ru2 inclusive are
non-degenerate, there exists a subset of non-degenerate geodesics of G which cor¬
respond in a one-to-one manner to the principal ellipses on Em(a) in such a fashion
that corresponding geodesics have the same index.
To establish (1) we identify Em( 0) with R' and R with R(' . The set of cycles
X-; (0) on 12' will have the great circles on Em{ 0) as a minimal set of closed
geodesics. The set of cycles on 12" corresponding to the cycles X j- i (0) after
partition will determine a minimal set G of closed geodesics on R. By virtue of
the preceding lemma the geodesics of G will have lengths between 2khi and
27t/u2 inclusive.
To establish (2) recall that the number of cycles X) ; (0) of dimension k equals
the number of principal ellipses of index k on Em(a), provided the constants
354
SOLUTION OF THE POINCARE CONTINUATION PROBLEM [ IX ]
ai > • • > am+ 1 are sufficiently near unity. Statement (2) follows from
Theorem 6.5 of Ch. VIII.
To verify statement (3) recall that the kth type number sum Mk of a set of
non-degenerate geodesics is the number of these geodesics with index k. Under
the hypotheses of (3) the geodesics of the set G are non-degenerate. It follows
from statement (2) that the number Mk of geodesics with index k in G is at least
as great as the number of principal ellipses on Em(a) with index k. The one-to-
one correspondence affirmed to exist in (3) can accordingly be set up as stated.
We also note the following. When g2 < 2/zi, none of the geodesics whose
existence is affirmed in the theorem can cover any other such geodesics an
integral number of times.
Continuation theorems
10. We shall conclude with two theorems on the analytic continuation of
closed geodesics. With Poincare the theory of the continuation of closed
geodesics was used to establish the existence of the basic geodesics. For us the
existence of the basic geodesics has been established by other means. The
theory of their continuation serves to describe their variation and the variation
of their type numbers with variation of the manifold.
We start with an analytic Riemannian m-manifold R , given in the large as
previously. We suppose that R is the initial member R0 of a 1 -parameter
family Ra of homeomorphic Riemannian manifolds depending on a parameter a
which varies on the interval
(10.1) Ogagl.
Let P o be any point on R. Let (x) be any admissible coordinate system repre¬
senting R neighboring P0, with (x) = (a) corresponding to P o. We represent the
point on Ra which corresponds to the point (x) on R by these same coordinates
(x), and suppose that the differential of arc on Ra takes the form
ds 2 = gn(x} a^dx'dx1,
where the coefficients ga(x, a) are analytic in the variables (x) and a for (x) near
(a) and a any number on the interval (10.1).
Let g' and g" be two closed curves on R. Let k represent a homeomorphism
between gf and g ". Let Dk be the minimum of the geodesic distances between
points of g' and g,f which correspond under k. Let d(g', g ") be the greatest lower
bound of the numbers Dk for all homeomorphisms k between g' and g The
number d(g', g") will be called the distance between g' and g " on R. Cf.
L rochet [1].
Suppose now that gf and g " lie on Ra> and Ra* respectively. Let yr be the
homeomorph of g" on Ra> and 7" the homeomorph of g' on Ra *. Of the two
numbers
d(g',y'), d(g",y")
[10]
CONTINUATION THEOREMS
355
on Ra' and Ra •, let 8 be the minimum. We define the distance between gr and
g " as the number
d(g', g") = [(«' - a'T +
Let Ha be a set of closed curves on Ra defined for all values of a ^ a0 suffi¬
ciently near a0. Let II be a set of closed curves on Rao . The set II a will be said
to tend to II as a limiting set as a tends to <*0 if for | a — a0 ] sufficiently small
each closed curve of II a (or II) is within an arbitrarily small distance of some
closed curve of II (or H a) respectively.
Let c be the length of a closed geodesic on Rao . Let J0 be the integral of arc
length on Rao. Let a and b be ordinary values of J0 which separate c from other
critical values of .70; a < c < b. Let Ha be the set of all closed geodesics on Ra
with lengths between a and b. We state the following:
(A). As a lends to ao, the set II „ of closed geodesics on Ra tends to a subset of
on R a0 as a limiting set.
We observe that the set, II a may be vacuous when a ^ a0. The subset of
Hao is then the null set. The proof of statement (A) is contained in the analysis
of critical sets of closed extremals in §3 of Ch. VIII. For our present purposes
the parameter a must be added to the variables employed in Ch. VITI.
Statement (A) contains no affirmation concerning the continued existence of
critical sets of closed geodesics on Ra as a is varied. The following theorem
makes such an affirmation and describes t he variation of critical sets of geodesics
with reference to their type numbers. In this theorem a finite ensemble of
critical sets of closed extremals will be termed a composite set of closed extremals.
First Continuation Theorem. Let K be a critical set of closed geodesics on
Rat- For a sufficiently near ao and not ao, there exists a composite, set Ka of closed
geodesics on Ra which tends to a subset of K as a tends to ao, and which possesses n
kth type number sum at least as great as that of K (k = 0,1, • • • ).
The set K„ is null at most when the type numbers of K are null.
To prove this theorem we regard the integral
Ja
a)
d*d^Ydt
at dt )
as a functional on R. The length of a curve g on Ra is given by the value of Ja
along the homeomorph of g on R. The geodesics on Ra will be represented by
the extremals of Ja on R. The preceding theorem is equivalent to the following
lemma concerning the functional Ja on R.
Lemma. Let G be a critical set of closed extremals belonging to the functional Ja„
on R. For a ^ a0 and sufficiently near «0 there exists a composite set Ga of closed
extremals belonging to Ja which tends to a subset of G as a tends to ao and which
possesses a kth type number sum at least as great as that of G.
The lemma will be made to depend upon the corresponding statement in IV of
356
SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
§2, Ch. VI concerning a function <i> of a parameter p and of a point P on a Rie-
manniann manifold R.
Let p be a positive constant uniformly limiting the lengths of admissible
elementary extremals determined by Ja on R for values of a near a0. Let c
be the value of Jao on the extremals of G , and p be an integer so large that
pp > c.
Let Rp be the Riemannian manifold of points (x) with p vertices on R. As
previously, we assign Rv an element of arc ds whose square is the sum of the
squares of the elements of arc of the respective vertices of a point (x) on Rp .
Let <t be the set of J-normal points (x) on Rp which belong to JO0 and are
determined by the extremals of G. Neighboring a let 2a be the set of all
J-normal points (x) on Rp which belong to Ja. The analysis of J-normal points
(x) in §7, Ch. VII, shows that for a sufficiently near <xo, and among points
sufficiently near <r, 2a forms a regular, analytic, Riemannian submanifold of Rp.
More precisely, let (x0) be any point of a, and (x) a set of pm coordinates locally
representing a neighborhood of (x0) on Rp. There then exists a set of parameters
(u) such that the points on 2a neighboring (x0) can be represented in the form
(10.2) x{ — a) (i = 1, • • • , mp).
The values of a in (10.2) are restricted to values near a0 and the sets ( u ) to sets
near the set (no) which determines (x0) on 2Qo. Let r be the number of variables
( u ). The functions y?*(w, a) are analytic in their arguments and possess a matrix
of first partial derivatives with respect to the variables ( u ) which has a rank r
when (u) = (uo) and a = a0.
The projection of on 2ao. We here make a digression in which we show that
2a can be projected onto 2^ neighboring a by means of geodesics on Rp orthog¬
onal to 2ao. For a sufficiently near «0 this will lead to an analytic home-
omorphism between 2a and 2«0 at least if 2a be restricted to points which
project into points on 2^ sufficiently near a. We shall obtain a representation
of this homeomorphism.
By using the power series representation of geodesics common in Riemannian
geometry, one can set up a non-singular analytic transformation of local co¬
ordinates (x) on Rp , into coordinates (y) of such a nature that 2ao is represented
in the space ( y ) by the coordinate r-plane f of the first r axes y\ and such that
the geodesics orthogonal to f are represented near (x0) by the set of straight lines
orthogonal to f in the space {y).
In terms of the coordinates ( y ) of Rp and the parameters (a) in (10.2) 2tt will
have a regular, analytic representation near (x0) of the form
(10.3) yi = ^(u, a) (i = 1, • • ■ , mp).
Since this representation reduces to the coordinate r-plane f when a = a0, we
have
0 s yp{(u, a0)
(t = r + 1, • • • , mp).
110]
CONTINUATION THEOREMS
357
The point ( y ) on 2„ which is given by (10.3) will project orthogonally into the
point
( y ) = [V(u> a), • * • , 1 pr(u, a), 0, • • • , 0]
on {*. We regard (?/, * • • , yr) as a set of parameters representing the point
[ y\ ■ * ■ , 0, • • - , 0]
on 2tt0. The relations between the parameters (u) which represent a point Q on
and the parameters (y) which represent the projection of Q on 2ao will take
the form
V1 = a) (i = 1, • • • , r).
These relations have the property that for a — a0,
flfrfrS ... 9}f,r)
D(uly , w') *
as follows from the regularity of the representation of 2ao.
Thus the relation between and its orthogonal projection on £ao is non¬
singular, analytic, and one-to-one provided a be sufficiently small and Xa be
restricted to points whose projections lie sufficiently near a.
We now return to a proof of the theorem.
Let P be any point on 2ao near a and (7 r) the point on 2a which projects
orthogonally into P. Let
HP, a)
be the value of Ja along the broken extremal g( w) which belongs to Ja. In
terms of coordinates (x) locally representing the point P on neighboring an
arbitrary point PQ of wao, the function will become a function F(x , a) which is
analytic in its arguments (x) and a of the nature of the function F(x, /1) of IV,
§2, Ch. VI.
When a ~ «o, the function possesses the critical set a. For values of a 9* a0
sufficiently near a{) the results of Ch. VI show that 4> will possess a set <ra of critical
points neighboring a writh a kth type number sum at least as great as that of a.
This statement is the basis of the proof of the theorem.
We now return to 2a and the functional Ja on R. Let a'a be the set of points
on 2a which projects on Rp into aa on 2ao. The set a'a is composed of critical
sets of /-normal points (7r) on 2a belonging to the functional J a on 2«. The
kth type number sum of aa will equal that of <r«, and hence be at least as great as
that of a. Let Ga be the set of closed extremals on R, belonging to /« and
determined by points (x) on <j'a . By virtue of the relations between critical sets
of /-normal points and the corresponding sets of closed extremals, the kth type
number sum of Ga will equal that of a'a, and hence be at least as great as that of a,
or at least as great as that of G.
358 SOLUTION OF THE POINCARfi CONTINUATION PROBLEM [ IX ]
The set Ga is the set of closed extremals whose existence is affirmed in the
lemma.
The proof of the First Continuation Theorem is now complete.
We return to the homeomorphic manifolds Ra, 0 ^ a ^ 1. Let fta be the
space ft defined by the integral of arc length on Ra . Let A be a &-cycle on ft0
which is not ft-homologous to zero on ft0. As we have seen there will exist a cycle
A„ on fta which “corresponds to A after partition. ” The cycle A* will not be
ft-homologous to zero on fta.
We now state a second theorem:
Second Continuation Theorem. For each value of a on the interval 1
there exists a minimal set Ma of closed geodesics on Ra belonging to the k-cycle A*.
A.'fa tends to a particular value a0, Ma tends to Mao The kth type number of Ma
is at least one.
A proper discussion of this theorem and its implications is beyond the scope of
these Lectures. A suitable treatment will be given elsewhere.
As an example we suppose that Ra is a family of analytic m-manifolds which
for a = 0 reduces to the m-ellipsoid Em(a) of Theorem 9.3. We take A as one of
the cycles A The corresponding ellipse g\3(a) on Em(a) will thus “con¬
tinue” into the set Ma in the sense of the preceding theorem. The set M a never
fails to exist and is uniquely defined.
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INDEX
Numbers refer to the appropriate pages
Abel, 313.
Alexander, vii, 146, 167, 182, 350.
Alcxandroff, 146, 359.
Bchaghel, 107, 122.
Bieberbach, 350.
Birkhoff, vi, 142, 143, 192, 305, 306, 307, 350.
Bliss, vi, 1, 3, 7, 8, 18, 10, 36, 64 , 80, 107,
113, 350.
Boeher, v, 83, 06, 07, 110, 113, 360.
Bolza, vi, 1, 19, 36, 46, 64, 80, 113, 192, 313,
360.
Boyee, 360.
Brown, 167, 108, 360.
Cairns, 360.
Carath6odory, 4, 18, 79, 192, 360.
Chicago Theses, 18, 360.
Cole, vii.
Cope, 36, 360.
Courant, 36, 62, 80, 145, 360.
Currier, 18, 64, 361.
Davis, 81, 361.
Dickson, 32, 361.
Dresden, 361.
Du Bois-Reymond, 2.
Eisenhart, 111, 152, 361.
Erdmann, 3.
von Escherich, 46, 52, 103.
Ettlinger, v, 96, 361.
Euler, 1, 19, 113.
Fr6chet, 209, 207, 354, 361.
Gergen, 361.
Graves, 4, 361.
Hadamard, vi, 1, 79, 113, 361.
Hahn, 61, 361.
Hausdorff, 361.
Hedlund, 79, 307, 361.
Hestenes, 361.
Hickson, 80, 361.
Hilbert, v, 3, 13, 36, 62, 80, 119, 120, 192,
361.
Hopf, 144, 361.
Hu, Kuen-Sen, 80, 362.
Ince, 05, 06, 362.
Jackson, 362.
Jacobi, 7, 8, 10, 16, 107, 120, 122, 125, 313.
John, 145, 362.
Johnson, 362.
Jordan, 172.
Kellogg, 178, 362.
v. Ker6kjd,rto, 362.
Kiang, Tsai-Han, 142, 362.
Kneser, 362.
Koopman, 108, 362.
Kronoeker, 9, 32, 145, 362.
Lagrange, vi, 36, 52, 80, 110, 173.
Larew, 362.
Lefschetz, vii, 107, 144, 146, 182, 252, 362.
Legendre, 5, 6, 16, 113, 114, 118, 120.
Lewis, 362.
Lindenbaum, 207, 362.
Liouville, 20, 102.
Littauer, 362.
Lusternik, 305, 307, 362.
McShane, 192, 363.
Mason, 3, 363.
Mayer, A., 11, 13, 120.
Mayer, W., 363.
Monger, 299, 363.
Morse, 18, 28, 36, 37, 45, 47, 61, 62, 64, 75,
78, 80, 99, 104, 110, 143, 145, 163, 180,
191, 305, 363.
Murnaghan, 364.
Myers, vii, 18, 28, 364.
Osgood, 19, 198, 257, 364.
Pitcher, 364.
Plancherel, 80, 364.
367
368
INDEX
PoincarS, v, vi, 19, 79, 143, 305, 300, 354,
364.
Price, 364.
Radon, 364.
Reid, 364.
Richardson, M., 191.
Richardson, R. D. G., 80, 364.
Richmond, 192, 364.
Rozenberg, 61, 365.
van Schaack, 143, 145, 365.
Schnirrelmann, 305, 307, 365.
Schoenberg, 365.
Signorini, 192, 365.
Smith, 191, 365.
Struik, 365.
Sturm, 20, 78, 80, 102, 365.
Tonelli, 192, 365.
Tucker, vii, 131, 146, 365.
Vebler., vii, 107, 110, 167 182, 365.
Vol terra, 365,
van der Waerden, 365
Walsh, 365.
Weierstraas, 3, 5, 15, 10, 112, 113, 114, 120.
White, 365.
Whitehead, 110, 365.
Whyburn, 145, 366.
Wintner, 366.
COLLOQUIUM PUBLICATIONS
1. H. S. White , Linear Systems of Curves on Algebraic Surfaces; F. S . Woods ,
Forms of Non-Euclidean Space; E. B. Van Vlecky Selected Topics in
the Theory of Divergent series and of Continued Fractions. 1905. 12
+ 187 pp. $2.75.
2. E. H. Moore , Introduction to a Form of General Analysis; E. J. Wilezyn-
skij Projective Differential Geometry; Max Mason , Selected Topics in
the Theory of Boundary Value Problems of Differential Equations.
1910. 10 + 222 pp. (Published by the Yale University Press.)
Out of print.
31. G. A. Bliss , Fundamental Existenr Theorems. 1913. Reprinted 1934.
2 + 107 pp. $2.00.
3 1 1. Edward Kasner , Differential-Geometric Aspects of Dynamics. 1913.
Reprinted 1934. 2 + 117 pp. $2.00.
4. L. E. Dickson f On Invariants and the Theory of Numbers; W . F. Osgood ,
Topics in the Theory of Functions of Several Complex Variables.
1914. 12 + 230 pp. $2.50.
51. G. C. Evans , Functionals and their Applications. Selected Topics, in¬
cluding Integral Equations. 1918. 12 + 136 pp. Out of print.
51 1. Oswald Veblen , Analysis Situs. Second edition. 1931. 10 + 194 pp.
$2.00.
6. G. C. Evans , The Logarithmic Potential. Discontinuous Dirichlet and
Neumann Problems. 1927. 8 + 150 pp. $2.00.
7. E. T. Belly Algebraic Arithmetic. 1927. 4 + 180 pp. $2.50.
8. L. P. Eisenharty Non-Riemannian Geometry. 1927. 8 + 184 pp.
$2.50.
9. G, D. Birkhoffy Dynamical Systems. 1927. 8 + 296 pp. $3.00.
10. A. B . Coble7 Algebraic Geometry and Theta Functions. 1929. 8 + 282
pp. $3.00.
11. Dunham Jackson , The Theory of Approximation. 1930. 8 + 178 pp.
$2.50.
12. Solomon LefschetZy Topology. 1930. 10 + 410 pp. $4.50.
13. R. L. Moore} Foundations of Point Set Theory. 1932. 8 + 486 pp.
$5.00.
14. J. F. Ritty Differential Equations from the Algebraic Standpoint. 1932.
10 + 172 pp. $2.50.
15. M. H. Stonef Linear Transformations in Hilbert Space and their Appli¬
cations to Analysis. 1932. 8 + 622 pp. $6.50.
16. G. A. Blissy Algebraic Functions. 1933. 9 + 218 pp. $3.00.
17. J. H. M. Wedderburn , Lectures on Matrices. 1934. 200 pp. $3.00.
18. Marston Morse , The Calculus of Variations in the Large. 1934. 380
pp. $4.50.
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